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Sufficient conditions for optimal control and the generalized problem of Bolza Zeidan, Vera Michel 1982

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SUFFICIENT CONDITIONS FOR OPTIMAL CONTROL AND THE GENERALIZED PROBLEM OF BOLZA by VERA MICHEL ZEIDAN B.Sc, Ecole Normale Superieure, Beirut 1976 B.Sc, Faculty des Sciences, Beirut 1976 M.A., Dalhousie University, 1978 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May 1982 0 Vera Michel Zeidan, 1982 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e head o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f CL The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 DE-6 (2/79) Abstract We develop i n t h i s t h e s i s four s u f f i c i e n c y c r i t e r i a f o r the generalized problem of Bolza. These r e s u l t s represent a u n i f i c a t i o n , i n the sense that they can be applied to both the calculus of v a r i a t i o n s and to optimal c o n t r o l problems, as well as to problems with nonsmooth data. The f i r s t c r i t e r i o n , "point convexity", extends the convexity approach of Rockafellar. However, we derive a "Hamiltonian-Jacobi" approach which can be applied when the point convexity assumption f a i l s to be s a t i s f i e d . The method employed f o r t h i s c r i t e r i o n brings to l i g h t a new point of view concerning the Jacobi condition i n the c l a s s i c a l calculus of v a r i a t i o n s . The l a t t e r can be considered as a condition which guarantees the existence of a canonical transformation transforming the o r i g i n a l Hamiltonian to a l o c a l l y concave-convex Hamiltonian. The t h i r d s u f f i c i e n c y c r i t e r i o n i s an extension of the Hamilton-Jacobi approach from optimal c o n t r o l to the generalized problem of Bolza. This r e s u l t gives r i s e to another s u f f i c i e n c y c r i t e r i o n i n terms of a c e r t a i n i n e q u a l i t y . Our theorems on s u f f i c i e n t conditions are c l o s e l y r e l a t e d . We prove that under c e r t a i n assumptions the l a s t three approaches can be u n i f i e d . By t h i s we mean that t h e i r hypotheses are equivalent. However, the point convexity, and hence the convexity, c r i t e r i o n turns out to have the most r e s t r i c t i v e hypotheses of the four. The generality of the theorems proven stems t o a great extent from the f a c t that not only non-differentiable but even i n f i n i t e -i i i valued functions are allowed i n the treatment. The u t i l i t y of using such functions appears when we apply these theorems to optimal c o n t r o l problems. TO Mimi, Hoda and Walter V Table of Contents Page Abstract i i L i s t of Figures v Acknowledgement v i Introduction 1 I. Survey of Known Results 7 1.1 The Three Problems 7 1.2 Necessary and S u f f i c i e n t Conditions f o r the Calculus of Variations Problems 14 1.3 Necessary and S u f f i c i e n t Conditions for Optimal Control Problems 17 1.4 Necessary and S u f f i c i e n t Conditions for the Generalized Problems of Bolza 21 I I . Convexity 27 I I I . Conjugacy 36 3.1 Statement of the S u f f i c i e n t Conditions 37 3.2 Concave-Convex Hamiltonians. A C r i t e r i o n f o r Local Concavity 43 3.3 Canonical Transformations 49 3.4 Proof of Theorem (5) 59 IV. A Hamilton-Jacobi Approach 76 V. I n t e r r e l a t i o n s h i p s Between the S u f f i c i e n t Conditions . . 85 VI. A p p l i c a t i o n to Optimal Control Problems 134 Bibliography 159 v i L i s t of Figures Page Figure (1) 85 v i i Acknowledgements The q u a l i t y of t h i s d i s s e r t a t i o n and the pleasure derived from w r i t i n g i t have been greatly enhanced by the i n t e l l e c t u a l i n t e r a c t i o n with Professor Frank Clarke. I am g r a t e f u l to him for his forbearance, support and p r o v i s i o n of subtle guidance throughout my endeavours. Working under Professor Clarke has been a r i c h experience thanks to h i s challenging ideas, sharp mind and f i n e sense of humour. I am indebted as well to Professor Haussmann for advice and encouragement. INTRODUCTION The main g o a l o f t h i s t h e s i s i s t o d e v e l o p s u f f i c i e n c y theorems f o r t h e g e n e r a l i z e d ^ p r o b l e m o f B o l z a . We r e q u i r e t h e s e c r i t e r i a t o u n i f y and e x t e n d known r e s u l t s f o r t h e c a l c u l u s o f v a r i a t i o n s , o p t i m a l c o n t r o l and g e n e r a l i z e d B o l z a p r o b l e m s . By t h i s we mean t h a t on t h e one hand, t h e s e c r i t e r i a a r e e q u i v a l e n t under c e r t a i n a s s u m p t i o n s , and on t h e o t h e r hand, t h e y a r e s u f f i c i e n t l y g e n e r a l t o encompass t h e t r a d i t i o n a l s e c o n d - o r d e r J a c o b i s u f f i c i e n t c o n d i t i o n i n t h e c l a s s i c a l c a l c u l u s o f v a r i a t i o n s , r e s u l t s o f R o c k a f e l l a r and C l a r k e on t h e "convex" g e n e r a l i z e d p r o b l e m o f B o l z a , and t h e H a m i l t o n - J a c o b i a p p r o a c h i n o p t i m a l c o n t r o l t h e o r y . The s t r e n g t h o f o u r " H a m i l t o n i a n - J a c o b i " a p p r o a c h i s i m p l i e d by t h e two f o l l o w i n g f a c t s . F i r s t , i t i s p o s s i b l e t o t r e a t t h e c a l c u l u s o f v a r i a t i o n s p r o b l e m s under l e s s r e s t r i c t i v e smoothness h y p o t h e s e s t h a n has so f a r been p o s s i b l e . Second, we a r e now a b l e t o o b t a i n a new s u f f i c i e n c y theorem f o r o p t i m a l c o n t r o l p r o b l e m s r e q u i r i n g l e s s r e s t r i c t i v e c o n d i t i o n s t h a n p r e v i o u s l y known. L e t us c o n s i d e r t h e f o l l o w i n g g e n e r a l i z e d p r o b l e m o f B o l z a : G i v e n two f u n c t i o n s L: TR.n x IR n * 3R u {+«>} and L: [a,b] x TB.n x 3R n —y 3R u {+ 0 0} , we seek t o m i n i m i z e £(x(a) , x ( b ) ) + L ( t , x ( t ) , x ( t ) ) d t o v e r a l l a b s o l u t e l y c o n t i n u o u s f u n c t i o n s x: [a,b] —y TR1 , w i t h d e r i v a t i v e x (a l m o s t e v e r y w h e r e ) . 2 The Hamiltonian of t h i s problem i s defined as follows: H(t,x,p) = sup{<p,v> - L ( t , x , v ) : v e E " } . In order that the problem be meaningful, we s h a l l always make the assumption that L i s measurable and H i s bounded by an integrable function. This implies that the i n t e g r a l i s defined (possibly +»). We also assume that neither £ nor L i s i d e n t i c a l l y + 0 0 . I f these assumptions are present, we say that the problem i s well-defined. The c l a s s i c a l calculus of v a r i a t i o n s problem i s a generalized problem of Bolza, where L i s a real-valued function, the boundary values x(a) = A and x(b) = B are given, and the minimum i s taken over a l l piecewise smooth x: [a,b] —y ]Rn . The optimal control problem i s defined as follows: Given functions f : [a,b] x iR n x 3Rm iR n , g : [a,b] x iR n x m m -> IR , 1°: IRn IR , and a subset U i n IR™ , seek to minimize b we £°(x(b)) + g ( t , x ( t ) , u ( t ) ) d t over a l l absolutely continuous x: [a,b] —> E." and measurable u: [a,b] ->- IRm s a t i s f y i n g x(t) = f ( t , x ( t ) , u ( t ) ) a.e., x(a) = A, and u(t) e U a.e. 3 In [22] Rockafellar proved that the generalized problem of Bolza subsumes both the calculus of v a r i a t i o n s and optimal c o n t r o l problems. Thus, any r e s u l t which holds f o r the generalized problem of Bolza can be t r a n s l a t e d i n terms of the other two problems. The existence of a s o l u t i o n for the generalized problem of Bolza was studied by Rockafellar i n [23] and necessary conditions were developed by Clarke i n [6] and [9]. However, there are two s u f f i c i e n c y c r i t e r i a f o r t h i s problem. The f i r s t one requires e i t h e r the functions -£(•,•) and L(t,',«) to be convex [20] and [21], or the Hamiltonian H(t,x,p) to be concave i n x and convex i n p [5]. The second involves a generalized Hamilton-Jacobi equation [16]. The primary objective of t h i s t h e s i s i s to develop s u f f i c i e n c y c r i t e r i a f o r the generalized problem of Bolza. Like most of the above a r t i c l e s t h i s t h e s i s focuses on the Hamiltonian H . Thus, our r e s u l t s complete the program of studying the generalized problem of Bolza from the point of view of the Hamiltonian. I t i s shown that the concavity assumption of [5] and the generalized Hamilton-Jacobi equation of [16] can be replaced r e s p e c t i v e l y by the "point concavity" and the "modified Hamilton-Jacobi" i n e q u a l i t y , which are weaker. The l a t t e r leads to a "new" type of s u f f i c i e n c y condition. Moreover, we develop s u f f i c i e n t conditions i n v o l v i n g a c e r t a i n i n e q u a l i t y c r i t e r i o n c a l l e d the "extended Jacobi" condition. This c r i t e r i o n guarantees the existence of a canonical transformation which takes the o r i g i n a l Hamiltonian (which i s not necessarily concave i n x) to a l o c a l l y concave-convex Hamiltonian. In the c l a s s i c a l case, we 4 show that our i n e q u a l i t y c r i t e r i o n i s i n f a c t equivalent to a w e l l -known condition i n v o l v i n g the Jacobi equation. Thus, t h i s equivalence sheds new l i g h t on the Jacobi condition, since the l a t t e r can now be interpreted as being e s s e n t i a l l y a necessary and s u f f i c i e n t condition for the existence of a canonical transformation f o r which the transformed Hamiltonian i s concave-convex. However, we ex h i b i t an example i n the c l a s s i c a l s e t t i n g where our form of the c r i t e r i o n i s easier to apply, and of course i t applies when others do not, such as i n c e r t a i n cases of nondifferentiable and/or extended real-valued data. By applying the c r i t e r i o n i n v o l v i n g the "extended Jacobi" condition to optimal co n t r o l we obtain a s u f f i c i e n c y theorem which can be applied to a large c l a s s of problems. Outline of the Thesis by Chapters Chapter I; We deal here e n t i r e l y with known r e s u l t s concerning the calculus of v a r i a t i o n s , optimal c o n t r o l and the generalized Bolza problems. We state these problems and we present the theorem [22] which shows the connection between the three problems. We then proceed to describe the s i t u a t i o n concerning necessary and s u f f i c i e n t conditions i n each of these problems. We present a well-known s u f f i c i e n c y theorem i n v o l v i n g the Jacobi condition i n the c l a s s i c a l calculus of v a r i a t i o n s , the Hamilton-Jacobi i n e q u a l i t y i n optimal c o n t r o l and the convexity approach f o r the generalized problem of Bolza. These r e s u l t s w i l l be extended i n the next chapters to more general s e t t i n g s . 5 Chapter I I : We derive here, for the generalized problem of Bolza, a s u f f i c i e n c y theorem r e q u i r i n g a "point convexity" assumption on the Hamiltonian. We then show that the s u f f i c i e n c y c r i t e r i o n of [5], which assumes that the Hamiltonian i s concave i n x , i s a s p e c i a l case of our r e s u l t . Chapter I I I : We develop here a second-order s u f f i c i e n c y c r i t e r i o n f or the generalized problem of Bolza, where the Hamiltonian 2 1 i s not assumed to be C , but rather C with L i p s c h i t z f i r s t d e r i v a t i v e s (C^ +) . Thus, the second-order condition i s a c e r t a i n i n e q u a l i t y stated i n terms of the generalized Jacobian of the L i p s c h i t z function H (t,-) . I t i s c a l l e d the "extended Jacobi" z condition. This i n e q u a l i t y guarantees the existence of a canonical transformation for which the transformed Hamiltonian i s concave-convex . We believe that the idea of employing canonical transformations f o r developing s u f f i c i e n c y c r i t e r i a has p o t e n t i a l applications beyond the domain of v a r i a t i o n a l problems. Chapter IV: We present here, f o r the generalized problem of Bolza, a s u f f i c i e n c y theorem i n v o l v i n g a "modified Hamilton-Jacobi" i n e q u a l i t y . As a c o r o l l a r y of t h i s theorem we obtain a s u f f i c i e n c y c r i t e r i o n which involves the generalized Hamilton-Jacobi i n e q u a l i t y . Thus, t h i s c o r o l l a r y extends the s u f f i c i e n c y c r i t e r i o n i n v o l v i n g the Hamilton-Jacobi i n e q u a l i t y from optimal c o n t r o l to the generalized problem of Bolza. Another consequence of our theorem here i s a "new" type of s u f f i c i e n t conditions, through which the connection between 6 the r e s u l t s here and the ones of Chapter I II w i l l be made. Chapter V: The entire chapter here i s a study of the re l a t i o n s h i p s between the d i f f e r e n t s u f f i c i e n c y c r i t e r i a developed i n the previous three chapters f o r the generalized problem of Bolza. We prove that i n the smooth case the "modified" and the generalized Hamilton-Jacobi i n e q u a l i t i e s are equivalent. Also, i t i s shown that under c e r t a i n conditions the Hamilton-Jacobi approach i s equivalent to the conjugacy approach of Chapter I I I . Furthermore, the "point convexity" approach turns out to be a s p e c i a l case of each of the others. F i n a l l y we show that i n the c l a s s i c a l s e t t i n g the extended Jacobi condition i s equivalent to the Jacobi condition. Chapter VI: The results of Chapters III and IV, are applied to obtain new s u f f i c i e n c y c r i t e r i a for optimal c o n t r o l problems. 7 CHAPTER I Survey of Known Results This thesis i s concerned with three types of problems: c l a s s i c a l c alculus of v a r i a t i o n s , optimal c o n t r o l , and generalized Bolza problems. The calculus of v a r i a t i o n s has a long h i s t o r y s t a r t i n g with the brachistochrone problem solved by the Ber n o u l l i s nearly 300 years ago. Its o r i g i n a l motivations came from c l a s s i c a l physics (mechanics, optics) and geometry. Since 1950 many new applications have been found and the theory of calculus of v a r i a t i o n s had to be extended. Some of these applications were to problems of i n d u s t r i a l process c o n t r o l and mathematical economics. These problems are not quite of the type considered i n the calculus of v a r i a t i o n s ; they are known as problems of optimal c o n t r o l . Recently, new applications have been found. As a r e s u l t , the theory of optimal c o n t r o l had to be extended, and a new problem, which i s now c a l l e d the generalized problem of Bolza, was introduced. In t h i s chapter, we w i l l f i r s t state these three problems and discuss how they are r e l a t e d . Then, i n the l a s t three sections we w i l l describe f o r each problem the s i t u a t i o n concerning necessary and s u f f i c i e n t conditions. 1.1 The Three Problems In t h i s section we present the calculus of v a r i a t i o n s , optimal c o n t r o l , and the generalized Bolza problems. We also show how these problems are r e l a t e d . 8 (a) C l a s s i c a l Calculus of V a r i a t i o n s : Suppose we are given a function L: [a,b] x TR1 x JR11 -> JR , and two constants A and n B i n TR . The calculus of v a r i a t i o n s problem i s : (P^) minimize (x) L(t , x ( t ) ,x(t) )dt over a l l piecewise smooth functions x: [a,b] -»• TR s a t i s f y i n g x(a) = A, x(b) = B For brevi t y , we c a l l a function x: [a,b] -*• TRn an arc i f i t i s piecewise smooth, and admissible f o r (P^) i f x(a) = A and x(b) = B. The Hamiltonian corresponding to the problem (P^) i s : (1.1) H ( t , X , p ) = sup{<p,V> - L ( t , x , v ) : V e TRn } . (b) Optimal Control Problems: Let f, g, £ be given functions such that r , . i i m _ n f : [a,b] x JR x JR —h JR g: [a,b] x jR n x jR m —> JR F Z : TRn —> TR Let U be a subset of TRm . The optimal control problem i s defined to be: 9 (C) minimize J(x,u) = -c Q(x(b)) + b g(t,x(t) ,u(t) )dt over a l l absolutely continuous functions x: [a,b] —> TRn with d e r i v a t i v e • m x (almost everywhere), and over a l l measurable functions u: [a,b] -*- IR s a t i s f y i n g x(t) = f (t,x(t) ,u(t)) a.e. , x(a) = A , u(t) e U a.e. The Hamiltonian of the problem (C) i s defined to be: (1.2) H(t,x,p) = sup{<p,f (t,x,u)> - g (t,x,u) : u e U l . The calculus of v a r i a t i o n s problem (P^) i s an optimal co n t r o l problem. In f a c t , i f we define u(t) = x(t) , the problem ( P ^ i s equivalent to the problem b m i n i m i z e a L ( t , x ( t ) , u ( t ) ) d t over a l l piecewise smooth x: [a,b] —*- ]Rn , and piecewise continuous functions u: [a,b] —> IRn s a t i s f y i n g 10 x(t) = u(t) x(a) = A, x(b) = B , which i s an optimal c o n t r o l problem with given boundary values. (c) The Generalized Problem of Bolza: Suppose we are given two functions £ and L such that L: [a,b] x JR" x ]Rn y JR U { + ° ° } £: IRn x jR n y JR u {+<*>} . The generalized problem of Bolza i s defined to be: (P) minimize J(x) = £(x(a),x(b)) + L ( t , x ( t ) , x ( t ) ) d t over a l l absolutely continuous functions x: [a,b] — y TRn , with d e r i v a t i v e x (almost everywhere). In t h i s context, such a function x i s c a l l e d an a r c . The Hamiltonian of the problem (P) i s defined by the "conjugate formula" (1.3) H(t,x,p) = sup{<p,v> - L(t,x,v) : v € TR11} . Because the functions L and £ may take the value + 0 0 , the problem (P) i s much more general than i t seems. I t has been shown that, under n o n r e s t r i c t i v e hypotheses, the optimal c o n t r o l problem (C) (which i s apparently d i f f e r e n t than (P)) can be placed within the 11 framework of the generalized problem of Bolza (P). To reformulate (C) we define: (1.4) £(x 1,x 2) = \ ( x 1 ) + £ Q ( x 2 ) , where Y ( x ) = i A 0 i f x = A + 0 0 i f x =j= A , and (1.5) L(t,x,v) = i n f {g (t,x,u) : v = f ( t , x , u ) , u e u} . The function L given by (1.5) i s equal to + °° i f the set on which the infimum i s taken happens to be empty. Such a function i s not only, no n d i f f e r e n t i a b l e , but i t can be discontinuous on the set where i t i s f i n i t e . Define the generalized problem of Bolza corresponding to the optimal c o n t r o l problem (C) by (P c) minimize J c ( x ) = <£(x (a) ,x(b)) + L ( t , x ( t ) , x ( t ) ) d t over a l l arcs x: [a,b] —»- 3Rn , where t and L are defined by (1.4) and (1.5) r e s p e c t i v e l y . From equations (1.3) and (1.5), the Hamiltonian corresponding to (P ) can be computed as follows: H(t,x,p) = sup{<p,v> - L( t , x , v ) : v e 3R } = sup{<p,v> - inf{ g ( t , x , u ) : v = f ( t , x , u ) , u e u}} v = sup{<p,f(t,x,u)> - g(t,x,u): ueu} . Thus, by using (1.2) we obtain that H(t,x,p) = H(t,x,p) whence, the problems (C) and (P c) have the same Hamiltonian. Now, define the functions K(t,x,v,u) = < g(t,x,u) i f u e U and v = f(t,x,u) + °° otherwise, and J(t,x,p,q) = sup{<p,v> + <q,u> - K(t,x,v,u) : (v,u) e 3Rn x3R n} = sup{<q,u> +<p,f(t,x,u)> - g(t,x,u): u eu} . D e f i n i t i o n ; The function J i s said to s a t i s f y the boundedness condition i f sup J(t,x,p,q) < <(>(t,r,p,q) , x I <r 13 where <j> i s a real-valued function on [a,b] x [o, +°°) x jR n x jR n such that <j> i s integrable i n t f o r f i x e d (r,p,q) . The following theorem which was proven by R.T. Rockafellar i n [22] expresses the connection between the problems (C) and (P^ ,) . A s i m i l a r r e s u l t appears i n [7]. Theorem 1 [22]: Assume that J s a t i s f i e s the boundedness condition. b Then, the i n t e g r a l J K ( t , x ( t ) , x ( t ) , u ( t ) ) d t i s well-defined (possibly a +°°) f o r any absolutely continuous function x and for any measurable function u , and i f i t s value i s not +°° , the function u i s summable. Furthermore, (a) L(t,x,v) defined by (1.5) i s lower semicontinuous i n (x,v), measurable i n (t,x,v), nowhere -°° , and the Hamiltonian H defined by (1.3) s a t i s f i e s the boundedness condition; (b) f o r every absolutely continuous x: [a,b] —• JRn , one has (where the minimum i s a t t a i n e d ) : (1.6) L ( t , x ( t ) , x ( t ) ) d t = min{ K ( t , x ( t ) , x ( t ) , u ( t ) ) d t : u measurable} Remark: Suppose that J s a t i s f i e s the boundedness condition. From equation (1.6), i t follows that, i f the p a i r (x,u) solves the problem (C), then x solves (P ) . Conversely, assume that x solves (P ) with f i n i t e optimal value. Then, L ( t , x ( t ) , x ( t ) ) , defined by (1.5), i s f i n i t e f o r t i n [a,b] a.e. Assume that the infimum i n (1.5) for L ( t , x ( t ) ,x(t)) i s attained at u(t). . By (1.6), the p a i r (x,u) 14 solves the problem (C). 1.2 Necessary and S u f f i c i e n t Conditions f o r the Calculus of Variations  Problem During the eighteenth and the nineteenth centuries, several s c i e n t i s t s focused t h e i r a ttention on the calculus of v a r i a t i o n s problem. Among them were Euler, Legendre, Lagrange, Hamilton, Jacobi, Weierstrass and Caratheodory. Their work led to several r e s u l t s con-cerning necessary and s u f f i c i e n t conditions. This section i s devoted to presenting some of these r e s u l t s . Recall that the calculus of v a r i a t i o n s problem i s : (P^) minimize J^(x) = b L ( t , x ( t ) , x ( t ) ) d t , over a l l piecewise smooth x s a t i s f y i n g x(a) = A , x(b) = B D e f i n i t i o n : An admissible arc x i s a weak l o c a l minimum for (P^) i f there e x i s t s a p o s i t i v e number e such that x minimizes J^(x) over a l l admissible arcs x s a t i s f y i n g x(t) - x ( t ) < e for a l l t e [a,b] , and x(t) - x ( t ) I < e f o r a l l t where x and x e x i s t 15 D e f i n i t i o n : An admissible arc x i s a strong l o c a l minimum f o r (P^) i f there e x i s t s a p o s i t i v e number e such that x minimizes (x) over a l l admissible arcs x s a t i s f y i n g | x ( t ) - x ( t ) | < e f o r a l l t e [a,b] It i s c l e a r that i f x i s a strong l o c a l minimum f o r (P^) i t i s also a weak l o c a l minimum. Now, we w i l l state some known necessary conditions f o r the problem (P^). For the proofs consult the. standard l i t e r a t u r e , e.g. [1], [11]. Suppose we are given an arc x . The notation <j) (t) w i l l mean that the function <j) i s evaluated at (t,x (t) ,x ( t ) ) ; and we g use " c l a s s i c a l notations" f o r d e r i v a t i v e s : i> = -—6 e t c . x 3x The Euler-Lagrange Equation: Assume that L(•,•,•) i s and that x i s a weak l o c a l minimum f o r (P^). Then, there e x i s t s a constant c such that L (t) = c + v L (s)ds x The Weierstrass Condition: Assume that L(•,*,•) i s C"*" and that x i s a strong l o c a l minimum f o r (P^) . Then, E(t,x(t) ,x(t) ,w) = L(t,x(t),w) - L(t) - <--w - x(t) ,L (t)> > 0 f o r a l l t e [a,b] and for a l l w e 3Rn . •16 2 The Legendre Condition: Assume that L (•,•,•) i s C and that x i s a weak l o c a l minimum for (P^) . Then L (t) > 0 w 3 Assume that L(•,•,•) i s C . The Jacobi equation i s defined by (1.7) (£ (t)h(t) + L (t)h(t)) - L (t)h(t) - £ (t)h(t) = 0 dt w vx xv xx D e f i n i t i o n : The point c i s said to be conjugate to a i f there i s a n o n t r i v i a l s o l u t i o n h to the Jacobi equation (1.7) which vanishes at a and also at c . The Jacobi Necessary Condition: Assume that L(•,*,•) i s 3 C and that x i s a weak l o c a l minimum f o r (P-^ ) • I f the strengthened Legendre condition holds, that i s i f L (t) > 0 f o r a l l t e [a,b] , w then there are no points conjugate to a i n (a,b). Concerning s u f f i c i e n c y i n the calculus of v a r i a t i o n s , a number of procedures are a v a i l a b l e to confirm the optimality of an arc s a t i s f y i n g the necessary conditions. These procedures involve the f i e l d of extremals, the Hamilton-Jacobi equation, and the Jacobi equation. Here, we only present the l a t t e r , because i t s extension to the generalized 17 problem of Bolza and to optimal control problems i s one of the main objectives of t h i s t h e s i s . 1 Theorem 2 [11]: Suppose that x' i s an admissible C -function. 3 Assume that L(•,•,•) i s C , and (1) (t) = L (t) , dt v x (2) L(t,x,w) - L(t,x,v) - <L v(t,x,v), w - v> > 0 for a l l w i n TRn , and for a l l (x,v) near (x,x), (3) L (t) > 0 , w (4) there are no conjugate points to a i n the i n t e r v a l (a,b]. Then, x i s a strong l o c a l minimum for the problem (P^) . Remark: Condition (1) i s the d i f f e r e n t i a l form of the Euler-Lagrange equation. Conditions (2), (3) and (4) are strengthened forms of the Weierstrass, Legendre and Jacobi conditions r e s p e c t i v e l y . Thus, the hypotheses of Theorem (2) are obtained by strengthening the necessary conditions i n a way that they become s u f f i c i e n t . 1.3 Necessary and S u f f i c i e n t Conditions f o r Optimal Control Problems Recall that the optimal c o n t r o l problem i s : (C) minimize J(x,u) = t (x(b)) + o b g(t,x(t) ,'u(t) )dt 18 over a l l absolutely continuous functions x: [a,b] -»• JR and measurable functions u: [a,b] JRM s a t i s f y i n g x(t) = f ( t , x ( t ) , u ( t ) ) a.e., x(a) = A , u(t) e U a.e. The necessary conditions known for the problem (C) are given by the maximum p r i n c i p l e of Pontryagin [17]. As we s h a l l soon see, these conditions extend some of the necessary conditions given i n section (1.2) f o r the calculus of v a r i a t i o n s problem. D e f i n i t i o n : A measurable function u from [a,b] to JRM i s said to be bounded i f the set u([a,b]) has a compact closure i n JRM . Assumptions: We assume that ZQ i s C^ " and the functions f(t,x,u) and g(t,x,u) are continuous i n (t,x,u) and continuously d i f f e r e n t i a b l e with respect to x on the d i r e c t product [a,b] x JRN x u . The Maximum P r i n c i p l e [17]: Let (x,u) be an admissible p a i r f o r the problem (C) such that u i s bounded. In order that (x,u) be optimal f o r (C) i t i s necessary that there e x i s t an arc p and a constant X ^ 0 such that p and X are not both zero, and (1) -p(t) = p ( t ) f (t,x(t) ,u(t)) - Xg(t,x(t) ,u(t)) a.e. , (2) max{<p(t),f(t,x(t),u)> - Xg(t,x(t),u): ue u} = < p ( t ) , f ( t , x ( t ) , u ( t ) ) > - Xg(t,x(t),u(t)) -19 for a l l t e [a,b] a.e., (3) -p(b) = I Q ( x ( b ) ) . x Remark: I f we consider the case where the minimum i n the problem (C) i s taken over a l l piecewise continuous functions u and piecewise smooth functions x such that |x(t) - x ( t ) | < 6 for some 6 > 0 . In t h i s case, i t i s shown i n [17] that the maximum p r i n c i p l e can be stated as above, but that (1) and (2) now hold f o r a l l t i n [a,b]. Now consider the calculus of v a r i a t i o n s problem (P^). We know that t h i s problem can be written as an optimal c o n t r o l problem where U = ZRn , f(t,x,u) E u , g(t,x,u) = L(t,x,u) . From the previous remark, i t follows that the maximum p r i n c i p l e can be applied to the problem (P.^ ) . Hence, i f L(•,•,*) i s C 1 the following holds: I f x i s a strong l o c a l minimum for (P^), there e x i s t a piecewise smooth function p and a constant X > 0 such that p and X are not both zero, and (i) p(t) = X L (t) , x ( i i ) max{<p(t),u> - XL(t,x(t),u): u e IRn} = <p(t),x(t)> - XL(t) . I f L i s c \ ( i i ) implies that 20 p(t) = AL (t) . V Using the f a c t that p and A are not both zero, the l a s t e q u a l ity above implies that A = 1 and hence, p(t) = L (t) . v Thus, from (i) we conclude that L (t) = p(a) + v L (s)ds , x which i s the Euler-Lagrange equation. Furthermore, since A = 1 and p(t)=L^(t) , equation ( i i ) implies that L ( t , x ( t ) ,x(t) +v) - L(t) > <v,L v(t)> f o r a l l v i n JRn and for a l l t e [a,b] . Thus the Weierstrass condition i s s a t i s f i e d , too. 2 -Now, i f we assume that L(*,«,*) i s C , equation ( i i ) y i e l d s that L (t) > 0 , w which i s the Legendre condition. Therefore, the Euler-Lagrange equation, the Weierstrass and Legendre conditions are extended to optimal c o n t r o l problems v i a the maximum p r i n c i p l e . The s i t u a t i o n concerning the s u f f i c i e n c y f or optimal 21 co n t r o l problems i s les s s a t i s f a c t o r y than i t i s for the c a l c u l u s of va r i a t i o n s problems. Most of the procedures known for the l a t t e r are unavailable f o r the former. However, we have some c r i t e r i a a v a i l a b l e i n s i m p l i f i e d contexts. For instance, when the d i f f e r e n t i a l equation i s l i n e a r i n both x and u , or when the candidate c o n t r o l function u i s assumed to l i e i n the i n t e r i o r of the con t r o l set U [15] . For the general case of optimal c o n t r o l problems we have only two c r i t e r i a . The f i r s t assumes that the Hamiltonian H , defined by (1.2), i s concave i n x and convex i n p [24] . This c r i t e r i o n , which i s also known for the generalized problem of Bolza, w i l l be presented f o r that problem i n the next s e c t i o n . The second c r i t e r i o n requires the existence of a continuously d i f f e r e n t i a b l e function <j)(t,x) solving the Hamilton-Jacobi i n e q u a l i t y <)>t(t,x) + H.(t,x,<fi (t,x)) < 0 , with equality at x(t) . As references see [2], [10], [14], [25], [26], and [27]. 1.4 Necessary and S u f f i c i e n t Conditions f o r the Generalized Problem  of Bolza Consider the generalized problem of Bolza given i n section (1.1): (P) minimize J(x) = £(x(a),x(b)) + over a l l arcs x: [a,b] —> TRU . L ( t , x ( t ) , x ( t ) ) d t , 22 The existence of a s o l u t i o n f o r t h i s problem (P) was studied by R.T. Rockafellar [23] . On the other hand, necessary conditions were developed by F.H. Clarke i n [7], [4], [6] and [9]. As we s h a l l soon see, these conditions are extensions to the generalized problem of Bolza of well-known necessary conditions i n the calculus of v a r i a t i o n s . D e f i n i t i o n [3]: Let C be a closed nonempty subset of TRn , and l e t c be a point i n C. The normal cone to C at c , denoted N Q ( C ) ' i s N (c) = ot co{lim s. (x. - c.)} , C . 1 1 1 where we consider a l l sequences of points (s^,x^,c_^) i n [O,00) x jR n x TRn such that x. converges to c , x. has c l o s e s t point c^ i n C , and the indicated l i m i t e x i s t s . D e f i n i t i o n [3] : Let f : JR n—• TR u {+°°} be a lower-semicontinuous function, and x be a point where f i s f i n i t e . The generalized gradient of f at x , denoted 3f (x) , i s 3f(x) = {peTR n: (p,-l) eN (x,f(x))} , where C i s the epigraph of f , i . e . , the set e p i f = { (s,r) e TRn x JR: f (s) < r} . Note: I f f i s l o c a l l y L i p s c h i t z , 3f i s the convex h u l l of a l l l i m i t s of the form 23 l i m Vf (x ) , where x^ converges to x and Vf(x^) e x i s t s f o r each i . Also, i f f i s C 1 , 9f(x) = {Vf(x)} . The Generalized Euler-Lagrange Equation [7]: Let x be a so l u t i o n to problem (P), where the functions £(•,•) and L(t,*,«) are lower-semicontinuous. I f some t e c h n i c a l hypotheses are s a t i s f i e d , then there e x i s t s an arc p such that (1.8) ( p ( t ) , p ( t ) ) e 9 L ( t , x ( t ) , x ( t ) ) a.e., (1.9) (p(a),-p(b)) e 3£(x(a) ,x(b)) , where 3L and denote the generalized gradients of L(t,*,«) and £(•,•) r e s p e c t i v e l y . From the note above i t follows that, i n the case where L i s C^, (1.8) reduces to the Euler-Lagrange equation i n d i f f e r e n t i a l form. Relation (1.9) i s a " t r a n s v e r s a l i t y condition". The Generalized Weierstrass Condition [4]: Let x be a s o l u t i o n for (P). I f some tec h n i c a l conditions are s a t i s f i e d , there e x i s t s for almost a l l t an element t, (t) of 3Rn such that f o r a l l v i n 3R L ( t , x ( t ) , x ( t ) +v) - L ( t , x ( t ) , x ( t ) ) > <v ,C(t)> . If L ( t , x ( t ) , * ) admits a gradient at x(t) , then 24 G(t) = V L ( t , x ( t ) , x ( t ) ) . I t i s c l e a r that, i n the calculus of v a r i a t i o n s case, t h i s condition reduces to the Weierstrass condition given i n section (1.2). The Hamiltonian Inclusions [6] and [9]: Let x solve the generalized problem of Bolza. Under some t e c h n i c a l hypotheses (which imply that H(t,*,«) i s L i p s c h i t z and £(•,•) i s lower semicontinuous) we can f i n d an arc p such that (1.10) (-p(t),x(t)) e 9H(t,x(t),p(t)) a.e., (p(a),-p(b)) e 8£(x(a),x(b)) , where 3H and 9£ are the generalized gradients of H(t,*,*) and £(•,•) r e s p e c t i v e l y . Relation (1.10) i s c a l l e d the "Hamiltonian i n c l u s i o n s " . In the calculus of v a r i a t i o n s r e l a t i o n (1.10) reduces to the Hamiltonian equations which, under some conditions, are equivalent to the Euler Lagrange equation. There are three known types of s u f f i c i e n t conditions f o r optimality i n the generalized problem of Bolza. The f i r s t assumes that there e x i s t s a unique function x s a t i s f y i n g the necessary conditions, and then uses the existence theorem of [23] to deduce the optimality of x . The second c r i t e r i o n stems from convex a n a l y s i s . It was developed by R.T. Rockafellar i n [20] and [21] , where the functions £ and L(t,«,«) are required to be convex. This r e s u l t i s given i n the following theorem. 25 Theorem [21]: Assume that £(•,•) and L(t,*,«) are convex. Let x , p be given arcs s a t i s f y i n g (a) (-p(t),x(t)) e 3H(t,x(t) ,p(t)) , (b) (p(a),-p(b)) e 3£(x(a) ,x(b)) . Then x solves the problem (P). Remark: From the d e f i n i t i o n of the Hamiltonian H given by (1 . 3 ) , and from the convexity of L(t,«,») i t follows that H(t,x,p) i s concave i n x and convex i n p . Another version of the s u f f i c i e n t conditions r e q u i r i n g convexity i s obtained by F.H. Clarke [5] f o r the case of f i x e d boundary values; x(a) = A and x(b) = B . In t h i s version the function L(t,*,«) i s not assumed to be convex, but instead, the Hamiltonian H i s required to be concave i n x . The t h i r d s u f f i c i e n c y theorem was obtained by D.C. O f f i n [16, Chapter 3] f o r the case where one boundary value i s f i x e d ; x(a) = A . This c r i t e r i o n requires the existence of a l o c a l l y L i p s c h i t z s o l u t i o n W(t,x) of the generalized Hamilton-Jacobi equation: max{a+H(t,x,B): (a,B) e 3W(t,x)} = 0 , where 3W i s the generalized gradient of W(#,*) . The objective of t h i s t h e s i s i s to develop s u f f i c i e n t conditions for the generalized problem of Bolza which extend and un i f y a l l the s u f f i c i e n t conditions presented i n t h i s chapter. 26 F i r s t , we generalize the convexity c r i t e r i o n ([20], [21] and [5]) to "point convexity". Then, we extend the s u f f i c i e n c y theorem in v o l v i n g the Hamilton-Jacobi i n e q u a l i t y to the generalized problem of Bolza. Also, we extend the Jacobi s u f f i c i e n t condition, i . e . , Theorem (2) to both optimal c o n t r o l and generalized Bolza problems. F i n a l l y , we show how these d i f f e r e n t conditions are r e l a t e d . 27 CHAPTER II Convexity D i f f e r e n t versions of s u f f i c i e n t conditions r e q u i r i n g some convexity assumptions can be found i n the l i t e r a t u r e . One version i s given i n [24] for optimal c o n t r o l problems. For the generalized problem of Bolza (P) , a s u f f i c i e n c y theorem i s developed i n [20] and [21] where the functions £(•,•) and L(t,*,*) are assumed to be convex. However, another version i s obtained i n [5] for the problem (P) with given boundary values: x(a) = A and x(b) = B . In t h i s l a t t e r the function L(t,«,') i s not assumed to be convex but instead the Hamiltonian H i s assumed to be concave i n x . In t h i s chapter we develop a s u f f i c i e n c y theorem for the generalized problem of Bolza where the functions -H(t,*,p) and £(•,•) are not required to be convex but must s a t i s f y a "point convexity" condition. We w i l l also show that a l o c a l version of the s u f f i c i e n c y theorem of [5], where the boundary values are not n e c e s s a r i l y f i x e d , i s a c o r o l l a r y of our theorem. The generalized problem of Bolza i s defined to be: (P) minimize J(x) = £(x(a),x(b)) + L ( t , x ( t ) , x ( t ) ) d t over a l l piecewise smooth functions x from [a,b] to 3Rn . We c a l l such a function x an arc. The functions £ and L take values i n IR u {+°°} . We assume that each of £ and L i s not i d e n t i c a l l y 28 {+°°} , otherwise any arc x i s optimal. The Hamiltonian of the problem i s defined by: H(t,x,p) = sup{<p,v> - L ( t , x , v ) : v e JRN} . D e f i n i t i o n : An arc x from [a,b] to TRN i s said to be admissible f o r the problem (P) i f we have £(x(a) ,x(b)) < + » . D e f i n i t i o n : An admissible arc x i s a strong l o c a l minimum f o r (P) i f there e x i s t s a p o s i t i v e number y such that x minimizes J(x) over a l l arcs x s a t i s f y i n g | x ( t ) - x ( t ) | < y for a l l t i n [a,b] . Suppose we are given arcs x , p and p o s i t i v e numbers e and <S . We define N(e , 6 ) = {(t,x,p): t e [a,b], |x - x(t) | < e , |p - p ( t ) | < 6} . Let L be the c o l l e c t i o n of Lebesgue measurable subsets of [a,b] and B the Borel subsets of 3RN x TRn . We denote by JL x B the a-algebra of subsets of [a,b] x JR11 x ]RN generated by products of sets i n L and B . The following n o n r e s t r i c t i v e assumption w i l l be made: (H^) L i s L x 8 measurable, and there e x i s t an integrable function p(') on [a,b] and p o s i t i v e numbers e, 6 such that f or (t,z) i n N(e , 6 ) 29 |H(t,z)| < p(t) . Theorem 3: Suppose we are given arcs x, p such that x i s admissible f o r (P). Assume that the hypothesis (H^) holds f o r some p o s i t i v e numbers e and 6 , and that (i) L ( t , x ( t ) , x ( t ) +v) - L(t, x ( t ) ,x(t)) > <p(t),v> for a l l v i n TRn and f o r almost a l l t i n [a,b] , ( i i ) there e x i s t s a p o s i t i v e number y such that f o r a l l x s a t i s f y i n g |x-x(t)| < y we have: H(t,x,j?(t)) - H(t,x(t) ,p(t)) < <-p(t),x - x(t)> a.e. , ( i i i ) f o r a l l x^, x^ such that | x 1 ~ x ( a ) | < y and , I x - x (b) | < y we have : £(x 1,x 2) - £(x(a),x(b)) > <p(a) ,x 1 - x(a)> - <p(b),x 2 - x(b)> . Then J(x) i s well-defined (possibly +°°) f o r x near x , J(x) i s f i n i t e , and x i s a strong l o c a l minimum f o r (P) . Remark 1: Condition (i) i s the necessary condition of Weierstrass given i n section (1.4) for the problem (P) where the function £ i s taken to be p . Remark 2: If the arc p also s a t i s f i e s the Hamiltonian i n c l u s i o n s and the t r a n s v e r s a l i t y condition (which are necessary), then conditions ( i i ) and ( i i i ) can be interpreted as being "point convexity" conditions 30 on -H(t,•,p(t)) and Zi',') . Remark 3 : As we s h a l l soon see, the hypothesis (H^) implies that J(x) i s well-defined (possibly +°°) f o r x near x . Proof of Theorem 3 : To prove that J(x) i s well-defined for x near x we consider x to be an arc s a t i s f y i n g |x(t) - x ( t ) | < e for a l l t i n [a,b] . Since L i s L x 8 measurable and since x and x are Lebesgue measurable (when x e x i s t s , i . e . , for almost a l l t i n [a,b]), then L ( t , x ( t ) , x ( t ) ) i s measurable i n t almost every-where. On the other hand, from hypothesis (H^) and the d e f i n i t i o n of the Hamiltonian i t follows that L ( t , x ( t ) , x ( t ) ) > <p(t),x(t)> - p(t) a.e. , and hence J(x) i s well-defined (possibly +°°) for x near x . Condition (i) and the d e f i n i t i o n of the Hamiltonian imply that (2.1) H(t,x(t),p(t)) = <p(t),x(t)> - L ( t , x ( t ) , x ( t ) ) a.e. By using the f a c t that x i s admissible and that H(t,x(t) ,p(t)) > - p (t) , where p ( ' ) i s integrable, equation (2.1) implies that J(x) i s f i n i t e . We assume without loss of generality that y < e , and we define: (2.2) H(t,x,p) = < H(t,x,p) i f | x - x ( t ) | < Y# p = p(t) +°° i f |x - x(t) | < y, p 4 P(t) -«> i f |x - x (t) | > Y and (2.3) 2(x l fx 2) = •( £(x ,x ) i f | x 1 - x ( a ) | < y, | x 2 - x ( b ) | < +°° otherwise. I t i s c l e a r that f or a l l x i n ]R and (x^,x 2) i n 3Rn xJRn the functions H and 2. s a t i s f y conditions ( i i ) and ( i r e s p e c t i v e l y . Consider the function (2.4) L(t,x,v) = sup{<p,v> - H(t,X,p) : p € TR } <p(t) ,v> - H(t,x,p(t)) i f |x-x(t)'| < Y +«> i f |x - x(t) | > Y Let us define the following problem: (P) minimize J(x) = £(x(a),x(b)) + L ( t , x ( t ) , x ( t ) ) d t , over a l l arcs x from [a,b] to TR Equations (2.3) and (2.4), and conditions ( i i ) and ( i i i ) 32 imply that f or any arc x we have: J(x) - J(x) > <p(a),x(a) - x(a)> - <p(b),x(b) - x(b)> a r + j (<p(t),x(t) - x(t)> +• b <p(t),x(t) - x(t)>}dt = 0 . Thus, x solves the problem (P) . On the other hand, f o r any x i n M s a t i s f y i n g | x - x ( t ) | < y equation (2.4) and the d e f i n i t i o n of H imply that: L(t,x,v) = <p(t),v> - H(t,x,p(t)) = i n f {<p (t) ,v - w> + L(t,x,w) : we ]Rn } < L(t,x,v) . So t h i s l a s t i n e q u a l i t y and (2.3) combined y i e l d : J(x) < J(x) , where x i s any arc s a t i s f y i n g |x(t) - x ( t ) | < y for a l l t i n [a,b] . But equations (2.1), (2.3), and (2.4) imply that 3 3 J(x) = £(x(a) ,x(b)) + = £(x(a),x(b)) + {<p(t),x(t)> - H(t , x ( t ) , p ( t ) ) } d t a b L ( t , x ( t ) , x ( t ) ) d t = J(x) . Since x solves (P) , i t follows that x minimizes J(x) over a l l arcs x s a t i s f y i n g |x(t) - x ( t ) | <y for a l l t i n [a,b] . Q.E.D. Remark: From the proof of Theorem (3) i t follows that, i f conditions ( i i ) and ( i i i ) are s a t i s f i e d everywhere, then J(x) i s w e l l -defined f o r a l l arcs x and x i s a global minimum for the problem (P). The r e s t of t h i s chapter i s devoted to d e r i v i n g from Theorem (3) another s u f f i c i e n c y theorem.for the problem (P) . In the case of f i x e d boundary values t h i s theorem i s i n f a c t a l o c a l version of the one given i n [5]. Suppose we are given arcs x and p . D e f i n i t i o n : The function H(t,«,p(t)) i s said to be l o c a l l y concave around x i f there e x i s t s a p o s i t i v e number y such that f o r any t i n [a,b] and for any x^, x^ s a t i s f y i n g |x ]_-x(t) | < y , |x - x ( t ) j < y 34 and f o r any A, 0 < A < 1 , we have: H(t,Xx + (1-A)x 2,p(t)) > XH(t,x ,p(t)) + (l-A)H(t,x 2,p(t)) . D e f i n i t i o n : A vector p i n 3Rn i s sa i d to be a subgradient (resp. supergradient) of a convex (resp. concave) function f : IRn ->- ]R at a point x i f , f o r a l l v i n JRU , f(x+v) - f(x) > <v,p> . (resp. <) D e f i n i t i o n : The set of a l l subgradients (resp. supergradients) of a convex (resp. concave) function f at a poin t x i s c a l l e d the s u b d i f f e r e n t i a l (resp. s u p e r d i f f e r e n t i a l ) of f at x and i s denoted by 3f(x) . Theorem 4: Suppose we are given arcs x , p such that x i s admissible f o r (P) . Assume that the hypothesis (H^) holds f o r some p o s i t i v e numbers e and 6 , and that H(t,*,p(t)) i s l o c a l l y concave on a y-neighbourhood of x and that £(x^,x 2) i s convex on { ( x 1 , x 2 ) : |x - x ( a ) | < y , | x 2 - x ( b ) | < Y > . Suppose that: (1) L ( t , x ( t ) ,x(t) + v) - L(t, x ( t ) ,x(t) ) ><p(t),v> fo r a l l v i n H n and for almost a l l t i n [a,b] , (2) (-p(t),x(t)) e 3H(t,x(t) ,p(t)) a.e., (3) (p(a),-p(b)) e 3£(x(a),x(b)) . 35 Then J(x) i s well-defined (possibly +°°) for x near x , J(x) i s f i n i t e , and x i s a strong l o c a l minimum for (P) . Remark 1: Since the Hamiltonian H i s defined as being the supremum of a f f i n e functions i n p , then H(t,x,«) i s convex, and hence 3H i n Theorem (4) denotes the product of the s u p e r d i f f e r e n t i a l at x(t) of the concave function H(t,«,p(t)) with the s u b d i f f e r e n t i a l at p(t) of the convex function H(t,x(t),») . Also 3-c (x (a) ,x (b)) i s the s u b d i f f e r e n t i a l at (x(a),x(b)) of the convex function £(•,•) . Remark 2: From [3, Proposition 1.2] we know that the generalized gradient of a convex function f i s the s u b d i f f e r e n t i a l of f . Thus, from section (1.4) we conclude that under some tec h n i c a l hypotheses conditions (2) and (3) and a weaker version of condition (1) are i n f a c t necessary. Remark 3: If the function L(t,x,v) i s convex i n v , then condition (2) and [19, Theorem 37.5] imply that condition (1) of Theorem (4) i s automatically s a t i s f i e d . Proof of Theorem 4: The proof consists of showing that the conditions of t h i s theorem imply that the conditions of Theorem (3) are s a t i s f i e d . Thus, we need to show that conditions ( i i ) and ( i i i ) are s a t i s f i e d . But t h i s follows from conditions (2) and (3) and from the f a c t that H(t,',p(t)) i s l o c a l l y concave near x and that £(•,•) i s l o c a l l y convex near (x(a),x(b)) . Q.E.D. 36 CHAPTER III Conjugacy The theory of conjugate points was employed to derive s u f f i c i e n t conditions f o r c l a s s i c a l c alculus of v a r i a t i o n s problems (Theorem 2). Some attempts have been made to develop s u f f i c i e n t conditions f o r optimal c o n t r o l problems using the same methodology. These attempts, however, were successful only i n s i m p l i f i e d contexts: f o r instance, when the con t r o l set U i s considered to be the whole space IR™ , or when the co n t r o l candidate u(t) i s assumed to be i n the i n t e r i o r of U (see D. Mayne [15]). In t h i s chapter we develop s u f f i c i e n t conditions f o r the generalized problem of Bolza which, as we s h a l l see i n chapter (5), turn out to be an extension of Theorem (2). When we apply these conditions to optimal c o n t r o l problems (see chapter 6 ) , we obtain a s u f f i c i e n c y theorem f o r the general case, that i s , when u(t) i s on the boundary of the con t r o l set U . Moreover, our conditions do not require the Hamiltonian H to be "point concave" i n x . How-ever, as we s h a l l show i n chapter (5), i f the Hamiltonian H i s smooth and "point concave", then these conditions are s a t i s f i e d . We r e c a l l that the generalized problem of Bolza i s : (P) minimize £(x(a),x(b)) + b r L ( t , x ( t ) ,x(t) )dt over a l l arcs x from [a,b] to 3RR . The Hamiltonian of the problem i s defined by: 37 H(t,x,p) = sup{<p,v> - L ( t , x , v ) : v e ]Rn } . In the. f i r s t section we present a s u f f i c i e n c y theorem f or the problem (P). Sections (3.2) and (3.3) are devoted to the study of a c r i t e r i o n f o r l o c a l concavity and of some properties of canonical transformations, which w i l l be our to o l s i n section (3.4) to prove the theorem. In t h i s chapter we continue to use the notations and notions of chapter (2) 3.1 Statement of the S u f f i c i e n t Conditions "> n Suppose we are given arcs x, p from [a,b] to 3R D e f i n i t i o n : The Hamiltonian H i s said to be C^~+ near (x,p) i f there e x i s t p o s i t i v e numbers e and 6 such that, f o r each t i n [a,b] , H(t,«) i s C"*" with l o c a l l y L i p s c h i t z f i r s t d e r i v a t i v e s on {z = (x,p) : | x - x ( t ) | < e, | p - p ( t ) | < 6} . If the Hamiltonian H i s C"'"+ then the gradient of H with respect to z , H (t,«) , i s l o c a l l y L i p s c h i t z and hence, the usual z n x n Jacobian matrix of p a r t i a l d e r i v a t i v e s D^H^CtjZ) e x i s t s f o r almost a l l z . D e f i n i t i o n : Let H be C^ + . The generalized Jacobian of H z(t,') at a point z , denoted S^H^ttjZ) , i s defined to be the convex h u l l of a l l matrices M of the form 38 M = lim {DnH__ ( t , z j } , where z^ converges to z and the usual Jacobian D^H^t,') e x i s t s at z. f o r each i . 1 We make the following assumptions: (H^) L i s L x 8 measurable, and there e x i s t an integrable function p(«) on [a,b] and p o s i t i v e numbers e and 6 such that, f o r (t,z) i n N(e ,6) H(t,z)I < p(t) , (H 2) the Hamiltonian H i s C 1 + near z = (x,p) with associated (e,6) , the function H (• ,z) i s continuous on [a,b] z for z near z , and the map (t,z) —»• 3 H (t,z) z z i s upper semicontinuous on N(e ,6) . By construction the generalized Jacobian 3 H (t,*) i s an z z upper semicontinuous multifunction on TRn x 3Rn ; hence i n the autonomous case, the hypothesis (H,,) simply reduces to saying that the map z ->• H(z) i s C"'"+ on {z = (x,p) : | x - x ( t ) | < e, |p-p(t)| < 6} . For n x n - matrices Q the notation Q > 0 (resp. Q > 0) means that Q i s p o s i t i v e d e f i n i t e (resp. semi-definite), and 39 Q1 > Q2 (resp. > Q2> means that Q± - Q2 > 0 (resp. Q± - Q2 > 0). D e f i n i t i o n : Let the Hamiltonian H be C"*"+ near z = (x,p) . We say that the extended Jacobi condition i s s a t i s f i e d at z i f there e x i s t s a L i p s c h i t z matrix function Q(') from [a,b] to the space of nxn-matrices such that, f o r a l l t i n [a,b], Q(t) i s symmetric and s a t i s f i e s : (3.1) n(t) -Q(t)y(t)Q(t) +Q(t)B(t) +<5(t)Q(t) - a (t) >0 f o r a l l t i n [a,b] , and for a l l matrices e 9 H ( t , z ( t ) ) , and for a l l n(t) e 9Q(t) . z z The following theorem represents a new s u f f i c i e n c y c r i t e r i o n f or l o c a l optimality of an arc x . Theorem 5: Suppose we are given arcs x, p such that x i s admissible f o r (p). Assume that, f o r some p o s i t i v e numbers e and 6 , the hypotheses (H^) and (H 2) are s a t i s f i e d , and (a) L ( t , x ( t ) , x ( t ) +v) - L ( t , x ( t ) , x ( t ) ) ><p(t),v> f or a l l v i n IRn and f o r almost a l l t i n [a,b] , (b) the arc z = (x,p) s a t i s f i e s J z(t) = H ( t , z ( t ) ) a.e. , a(t) <5(t) B(t) yit) 40 where J = 0 -I 0 with the n x n - i d e n t i t y matrix I I (c) the extended Jacobi condition i s s a t i s f i e d at z with associated matrix function Q(') , (d) there e x i s t s a p o s i t i v e number a such that, for a l l c and d Then J(x) i s well-defined (possibly +°°) f o r x near x , J(x) i s f i n i t e , and x provides a strong l o c a l minimum f o r (P). The proof of t h i s theorem w i l l be given i n section (3.4). Remark 1: From section (1.4) we know that, under some a d d i t i o n a l hypotheses, the generalized Weierstrass condition and the Hamiltonian Inclusions are necessary conditions. Thus, the s u f f i c i e n t conditions given by Theorem (5) are obtained by strengthening these necessary conditions. Remark 2: I f L(t,x,v) i s a closed convex function i n v then, from [19, Theorem 12.2], i t follows that L(t,x,v) = sup{<p,v> - H(t,x,p): p e JRn} . In t h i s case, condition (b) implies that condition (a) i s s a t i s f i e d . Also, the measurability of H implies the measurability of L and hence, i n t h i s case, a l l the hypotheses of Theorem (5) can be formulated i n terms of the Hamiltonian H . s a t i s f y i n g c < a and d < a we have (3.2) £(x(a) +c, x(b) +d) - £(x(a) ,x(b)) ><p(a),c> 41 Remark 3: If £(•,•) i s C near (x(a),x(b)) then, by using the fa c t that the t r a n s v e r s a l i t y condition of section (1.4) i s necessary, condition (d) of Theorem (5) i s s a t i s f i e d when we have: -Q(a) 0 0 Q(b) < V £(x(a),x(b)) Remark 4: Suppose we are given boundary values, i n the sense that the problem (P) reduces t o : b minimize L ( t , x ( t ) , x ( t ) ) d t subject to x(a) = A and x(b) = B . In t h i s case the function £(*,•) i s defined by £(x ,x 2) = 0 +00 i f x^ = A and = B elsewhere, and an admissible arc s a t i s f i e s x(a) = A and x(b) = B , so that £(x(a),x(b)) = 0 . In t h i s case, the i n e q u a l i t y (3.2) i s automatically s a t i s f i e d . In f a c t , then x i s admissible, so £(x(a),x(b)) = 0 . On the other hand, when c or d i s not zero, then the l e f t hand side of (3.2) i s + °° . Thus, i n t h i s case (3.2) holds f o r any c and d . 3 - 2 Concave-Convex Hamiltonians. A C r i t e r i o n f o r Local Concavity This section serves as a preparation f o r the proof of 4 2 Theorem ( 5 ) . As we s h a l l see i n section (3.4), the main idea of the proof of our theorem i s to f i n d a transformation taking the variables x, p and the Hamiltonian H(t,x,p), which i s not necess a r i l y concave i n x , to new v a r i a b l e s X , P and to a new Hamiltonian H*(t,X,P) , such that the following properties are s a t i s f i e d . I f X i s the transformed arc of x , then the new Hamiltonian i s convex i n P and l o c a l l y concave i n X around X . Also, i f X solves the transformed problem (P*) , then x solves the o r i g i n a l problem (P). Thus, i f such a transformation i s found, the proof of the theorem reduces to applying a s u f f i c i e n c y theorem stemming from convex analysis to the transformed problem (P*). Such a theorem w i l l be presented here, as well as a c r i t e r i o n which guarantees the concavity i n X of the transformed Hamiltonian H* . " n Suppose we are given arcs X , P from [a,b] to JR For given p o s i t i v e numbers e* and 6* we define N*(e*,6*) = { (t,X,P) : t e [a,b] , |x - X(t) | < e*, |p - P(t) | < <5*} , N * ( e * , ° o ) = {'(t,X,P) : t e [a,b] , ]x - X(t) | < e*, P e TRU } . Let H*(t,X,P) and Z*[X ,X^) be given functions defined re s p e c t i v e l y on N*(e*,°°) and on { ( X , X 2 ) : |x - X ( a ) | < £*, |x - X ( b ) | < e*} , where £* takes values i n JR u {+00} . 43 D e f i n i t i o n : The function H (t, • ,P(t)) i s said to be l o c a l l y concave around X i f there e x i s t s a p o s i t i v e number y* such that f o r any t i n [a,b] and for any X 2 s a t i s f y i n g 1^ - X ( t ) | < y*, X 2 - X ( t ) < y , fo r a l l X: 0 < X < 1 , we have: H*(t,XX 1+ (l-X)X 2,P(t)) > XH*(t,X ,p(t)) + (l-X)H*(t,X 2,P(t)) Define the function L by the conjugacy formula (3.4) L*(t,X,V) = sup{<P,V> - H*(t,X,P): P e / } , then L* i s defined on N*(e*,°°) . Consider now the following generalized problem of Bolza (P*) minimize J*(X) = £*(x(a) ,X (b)) + L ( t , X ( t ) , X ( t ) ) d t over a l l arcs X from [a,b] to 3Rn s a t i s f y i n g X(t) - X ( t ) | < e* f o r a l l t e [a,b] The following hypotheses w i l l be made: (H*) H* i s L x B measurable, and there e x i s t an integrable 44 * function p (•) on [a,b] and a p o s i t i v e number 6 * such that for (t,Z) i n N * ( e * , 5 * ) |H*(t,Z)| < p*(t) , (H*) f or t i n [a,b] the function H*(t,') i s l o c a l l y L i p s c h i t z on {Z=(X,P): | x - X ( t ) | < e*, | P - P ( t ) | < 6 * } . The following r e s u l t i s a s u f f i c i e n c y c r i t e r i o n f o r the problem (P*) based on convexity a n a l y s i s . Proposition 3.1: Suppose we are given arcs X, P such that X i s admissible f o r the problem (P*). Assume that the hypotheses- (H|) and (H|) hold, and that . (1) H*(t,«,P(t)) i s l o c a l l y concave around X and H*(t,X,«) i s convex f o r a l l (t,X) i n the set {(t,X) : t e [a,b] , |x - X(t) | < e*} ; (2) the arc Z = (X,P) s a t i s f i e s JZ(t) £ 3 H*(t,X(t),P(t)) a.e., Li where 3 H*(t,*) i s the generalized gradient of H*(t,«) ; 45 (3) the function t* s a t i s f i e s £ * ( X ( a ) + c*,X(b) +d*) - £*(X(a) ,X(b)) > <P(a),c*> - <p(b),d*> fo r a l l c* and d* ; |c*| < a and |d*| < a* , where a* i s some p o s i t i v e number. Then J*(X) i s well-defined (possibly +«>) f o r X near X , J*(X) i s f i n i t e , and X provides a strong l o c a l minimum f o r (P*). Proof: Hypothesis (H*) and condition (1) imply that H*(t,X,«) i s a proper closed convex function. Thus, from equation (3.4) and [19, Theorem 12.2] i t follows that the Hamiltonian of the problem (P*) i s H* . So, the proof w i l l proceed by showing that the conditions of Theorem (3) are s a t i s f i e d , where H and L are replaced by H* and L . We have to check that conditions (i) and ( i i ) of Theorem (3) hold. Condition (2) and the concavity of H*(t,*,P(t)) y i e l d that condition ( i i ) i s s a t i s f i e d . On the other hand, L*'(t,X,«) i s convex by d e f i n i t i o n and hence, conditions (1) and (2) and [19, Theorem 37.5] imply that condition (i) of Theorem (3) also i s s a t i s f i e d . Q.E.D. In the previous p r o p o s i t i o n we have assumed that the function H* (which w i l l l a t e r play the r o l e of the transformed Hamiltonian) i s l o c a l l y concave. The following r e s u l t establishes a c r i t e r i o n for l o c a l concavity of H* . In section (3.4) we w i l l show that the extended Jacobi condition, which i s condition (c) of Theorem (5), implies the hypotheses of t h i s c r i t e r i o n . 46 Given a p o s i t i v e number y* w e define N (y*) = {XeIR n: |x-X(t)| < y*} Proposition 3.2: Suppose that for some p o s i t i v e number y* and f o r 1+ each t i n [a,b] the function H* (t, • ,P (t)) i s C on N (y*)' . * * Assume i n addition that H (t,X,P (t)) < 0 whenever H e x i s t s on XX XX { ( t , X , P ( t ) ) : t e [ a , b ] , |x - X(t)| < y*}.:. Then H*(t,•,P(t)) i s l o c a l l y concave around X . Proof. We w i l l show that for each t i n [a,b] the function H*(t,•,P(t)) i s concave on any l i n e segment i n N (y*) . Consider t i n [a,b], X i n (y*) , and d i n o o t o TR ; d | 0 . Since N (y*) i s convex, then we can f i n d r e a l numbers o T and T 2 such that X q + i d e Nfc (y*) for x e (T ,T 2) , o and X + i d | N (y*) for x 4 (T n,T 0) . o t 1 2 Define g(x) = H*(t ,X + xd,P(t )) as a function from (T, ,T„) to O O O 1 2 TR . Since the function H*(t ,•,P(t )) i s C 1 + on N (y*) , g(») o i s also C on (T^,T 2) , and hence, by the chain rule i n [8, Section 13], we obtain: g(x) = H* (t ,X + xd,P(t ) ) . d , X o o o 47 and 3g(x) c d.3 H*(t , X + xd,P(t ))d , X X o o o where 3g denotes the generalized gradient of the function g , and * * * ~ 3 H i s the generalized Jacobian of H . Since H ( t , X , P ( t ) ) < 0 X X X X X tfc ^ /s whenever H e x i s t s on { ( t , X , P ( t ) ) : t e [a,b] , | X - X ( t ) | < y*} , f o r any ( t , X , P ( t ) ) belonging to the above set the following holds: * n for v e 3 H ( t , X , P ( t ) ) and for any d i n 3R X X d.vd < 0 . * Thus, for a l l x e (Tn ,T_) and f o r a l l v e 3„H„(t , X +id,P(t )) , 1 2 X X O O O d.vd < 0 . Therefore, the function g(') i s nonincreasing, hence g(*) i s concave on (T^,T 2) . Since the argument given above i s v a l i d for any choice of t Q i n [a,b], X q i n N (y*) , and d i n TRn , we conclude that H*(t,«,P(t)) i s l o c a l l y concave around X . Q . E . D . 3 . 3 Canonical Transformations In t h i s section we discuss the p a r t i c u l a r c l a s s of canonical transformations which i s the basis of the proof of Theorem (5) . As we s h a l l see s h o r t l y , t h i s c l a s s has very useful properties. Any 48 t r a n s f o r m a t i o n o f t h i s c l a s s t a k e s t h e o r i g i n a l H a m i l t o n i a n H(t,x,p) t o a new H a m i l t o n i a n H*(t,X,P) which i s a u t o m a t i c a l l y convex i n P . A l s o , i f t h e t r a n s f o r m e d a r c X s o l v e s t h e new problem ( P * ) , t h e n the o r i g i n a l a r c x would s o l v e the problem ( P ) . In s e c t i o n (3.4) we w i l l see t h a t t h e extended J a c o b i c o n d i t i o n g u a r a n t e e s t h e e x i s t e n c e o f one t r a n s f o r m a t i o n b e l o n g i n g t o t h i s c l a s s and g e n e r a t i n g a l o c a l l y concave H a m i l t o n i a n H*(t,',P) . Suppose we a r e g i v e n an a r c x , a p o s i t i v e number a , and two f u n c t i o n s h ( • ) , F ( • , • ) such t h a t h: [a,b] *• TRU F: { ( t , x ) : t e [ a , b ] , | x - x ( t ) i < a } >- TRn , where h i s C"^  , and F s a t i s f i e s t h e f o l l o w i n g c o n d i t i o n s : (A) F ( * , x ) i s L i p s c h i t z , f o r almost a l l t i n [a,b] the f u n c t i o n F ^ C t , ' ) e x i s t s and i t i s L i p s c h i t z , f o r some i n t e g r a b l e f u n c t i o n B(') on [a,b] we have | F t ( t , x ) | < B(t) , the f u n c t i o n 2 F ( t , * ) i s C and i n v e r t i b l e , and F^(-,x) i s L i p s c h i t z . L e t g(t,«) be the i n v e r s e f u n c t i o n o f F(t,«) ; g(t,«) = F 1 ( t , « ) , and d e f i n e (3.5) X ( t ) = F ( t , x ( t ) ) S i n c e F ( t , * ) i s c o n t i n u o u s , g ( t , * ) a l s o i s c o n t i n u o u s and hence, t h e r e e x i s t s a p o s i t i v e number a* such t h a t f o r a l l t i n [a,b] and f o r a l l X s a t i s f y i n g | x - X ( t ) j < a* we have 4 9 |g(t,X) - g(t,X(t) [ < a . Moreover, i t follows from the equation P(t,g(t,X)) = X that (3.6) g t(t,X) = - ( F x ( t , g ( t , X ) ) ) _ 1 F (t,g(t,X)) and (3.7) gv(t,X) = (F ( t , g ( t , X ) ) ) - 1 . Thus, using hypothesis (A) and equations (3.6) and (3.7) we conclude that g v(',X) and g(*,X) are L i p s c h i t z , g (t,«) i s C 1 , for A X almost a l l t i n [a,b] the function g(t,») e x i s t s and i s L i p s c h i t z , and there e x i s t s an integrable function £*(•) on [a,b] -such that |gt(t,X)| < B*(t) . Consider the canonical transformation whose generating function has the form: (3.8) ¥(t,X,p) = - (p - h(t)).g(t,X) , where (t,X,p) e { (t,X,p) : t e [a,b] , | x - X ( t ) | < a*, peffi"} . From the properties of g(*, #) and h(«) we have that Ht,',*) i s C 2 , and ¥ (t,*,*) e x i s t s f o r almost a l l t i n [a,b] , and i s L i p s c h i t z . By using arguments s i m i l a r to those used i n [11] for canonical transformations we obtain that the canonical transformation corresponding to the generating function ¥ transforms the o r i g i n a l 50 variables x , p and the o r i g i n a l Hamiltonian H(t,x,p) to new * variables X , P and to a new Hamiltonian H (t,X,P) i n the followinq o way: (3.9) x = - Y (t,X,p) = g(t,X) , (3.10) P = - ¥ x(t,X,p) = ( g x ( t , X ) ) T ( p - h ( t ) ) , T where A denotes the transpose of the matrix A , and (3.11) H*(t,X,P) = H(t,x,p) + Y (t,X,p) = H(t,x,p) + <h(t),g(t,X)> - <g t(t,X) , p -h(t)> , where x , p are considered as functions of (X,P) obtained from (3.9) and (3.10). We a l s o have * (3.12) PdX - H Q(t,X,P)dt = pdx - H(t,x,p)dt - d(<h(t) ,g(t,X)>) , f o r t e [a,b] a.e. Since <3^(t,') e x i s t s f or t e [a,b] a.e., our new Hamiltonian * H Q(t,X,P) i s defined for t e [a,b] a.e. We now define: 51 (3.13) H*(t,X,P) = i H*(t,X,P) whenever g t(t,*) exists otherwise ; then H*(t,X,P) is defined on N*(a*,°°) (for a l l t) . One of the advantages of using this particular canonical transformation i s that the transformed Hamiltonian H* is automatically convex in P . In fact, the function H(t,x,*) is convex by definition, and from (3.9) and (3.10) we have that p i s an affine function of P and that x is independent of P . Thus, (3.11) implies that H*(t,X,») is convex and hence, H*(t,X,«) is convex. Consider the following functions (3.14) L*(t,X,V) = sup{<P,V> - H*(t,X,P) : P e l " } , and (3.15) £*(X ,X ) = <h(b)fg(b,X )> - <h(a),g(a,X )> + £(g(a,X ),g(b,X2)) ; L* i s defined on N*(a*,°°) . Define now the transformed problem to be b L*(t,X(t),X(t))dt , (P*) minimize J*(X) = I*(X(a),X(b)) + over a l l arcs X: [a,b] IRn such that X(t) - X ( t ) | < a* f o r a l l t e [a,b] where X , L* and I* are defined by (3.5), (3.14) and (3.15) r e s p e c t i v e l y . The connection between the transformed problem (P*) and the o r i g i n a l problem (P) i s given by the following p r o p o s i t i o n . We continue to use the notations and notions of the previous sections. Proposition 3.3: Suppose that we are given arcs x, p such that x i s admissible f o r (P) . Assume that hypothesis (H^) holds, and (i) L ( t , x ( t ) ,x(t) + v) - L ( t , x ( t ) ,x(t)) ><p(t),v> fo r a l l v i n TRn and f o r almost a l l t i n [a,b] , ( i i ) H(t,«) i s l o c a l l y L i p s c h i t z near z = (x,p) , and that z s a t i s f i e s Jz(t) e 8 H(t,z(t)) a.e. Then (a) J(x) i s well-defined near x , and J(x) i s f i n i t e ; (b) there e x i s t s an arc P such that Z = (X,P) and H* s a t i s f y (H*), (H*) and the Hamiltonian i n c l u s i o n s ; (c) i f X i s a strong l o c a l minimum f o r (P*) then x i s a strong l o c a l minimum f o r (P) 53 Proof: Hypothesis (H^) and condition (i) imply that J(x) i s w e l l -defined near x and J(x) i s f i n i t e (see the proof of Theorem 3). The arcs x, p are transformed v i a equations (3.9) and (3.10) to X and P . So, the.function X i s given by (3.5) and (3.16) P(t) = ( g x ( t , X ( t ) ) ) T ( p ( t ) - h ( t ) ) . Since L i s L xB-measurable, H i s , too. On the other hand we 1 2 have that h i s C , g(t,-) i s C , g (•,X) and g(«,X) are X L i p s c h i t z , g t ( t , * ) e x i s t s and i s L i p s c h i t z f o r almost a l l t i n [a,b] , and |g t(t,X)| < 3*(t) where B* integrable. Thus, X and P are absolutely continuous. Moreover, from equations (3.11), (3.9) and (3.10), and from hypothesis (H^) i t follows that H q i s L x Im-measurable, H*(t,',«) i s L i p s c h i t z near (X,P) , and there e x i s t p o s i t i v e numbers e*, 6* , with e* < a* , and an integrable function p*(*) on [a,b] such that H*(t,X,P)| < p*(t) , o f o r a l l t where H*(t,-,«) e x i s t s and for a l l (X,P) : | x - X ( t ) | < e* o and | p - P ( t ) | < 6* . Therefore, the Hamiltonian H* s a t i s f i e s hypotheses (H*) and (H*) . The main property of canonical transformations i s that they conserve the "Hamiltonian i n c l u s i o n s " (see [16]). Thus, condition ( i i ) implies that the transformed arc Z = (X,P) s a t i s f i e s 54 JZ(t) e 9H*(t,Z(t)) a.e. o and hence, from (3.13), (3.17) JZ(t) e 3H*(t,Z(t)) a.e. To complete the proof, i t remains to show that condition (c) i s s a t i s f i e d . F i r s t we observe that the a d m i s s i b i l i t y of x for (P) implies that X i s admissible f o r (P*) . Furthermore, equations (3.14) and (3.17), and the convexity of H*(t,X,0 imply that (3.18) L * ( t , X ( t ) , X ( t ) ) =<P(t),X(t)> - H*(t,X(t),P(t)) a.e. and hence, J*(X) i s f i n i t e . Now, suppose that X i s a strong l o c a l minimum f o r (P*). This means that there e x i s t s a p o s i t i v e number y* (y* < e*) such that X minimizes J*(X) over a l l arcs X s a t i s f y i n g By using the f a c t that F(•,•) i s continuous, we can f i n d a p o s i t i v e number y(y < c) such that f o r any arc x s a t i s f y i n g X(t) - X(t) | < Y * fo r a l l t e [a,b] x(t) - x(t) | < Y f o r a l l t e [a,b] the arc X(t) = F ( t , x ( t ) ) s a t i s f i e s 55-|x(t) - X(t) I < y* • Let x be an arc such that |x(t) - x ( t ) j < y for a l l t i n [a,b], and l e t X(t) = F ( t , x ( t ) ) . By using (3.14), (3.13), (3.11), (3.9), (3.10) and the d e f i n i t i o n of the Hamiltonian H we obtain that, for almost a l l t i n [a,b] , L*( t , X ( t ) , X ( t ) ) = sup{<P,X(t)> - H*(t,X(t),P): P e IR } = sup{<P,X(t)> - H( t , g ( t , X ( t ) ) , ( g (t,X(t))) P+h(t)) X - <h(t) ,g(t,X(t) )> + <g (t,X(t) ) , (g (t,X(t) ) ) T P>:Pe]Rn} = sup i n f {<P,X(t)> - <v-g. ( t , X ( t ) ) , ( g v ( t , X ( t ) ) ) P> ' P v t X + L( t , g ( t , X ( t ) ) , v ) - <h(t),g(t,X(t))> - <h(t),v>} < i n f sup{<P,X(t) - (g ( t , X ( t ) ) ) _ 1 ( v - g (t,X(t)))> v P x r + L(t , g ( t , X ( t ) ) , v ) - <h(t),g(t,X(t))> - <h(t),v>} = i n f K(v) , v where K(v)= { |L(t,g(t,X(t)) ,v)-<h(t) ,g(t,X(t) )>-<h(t) ,v> i f v = g f c (t ,X (t))+g x (t ,X (t)) X (t) (+°° i f v4=g t(t,X(t))+g x(t,X(t))X(t) Since x(t) = g ( t , X ( t ) ) , where g(«,X) i s L i p s c h i t z , g(t,«) i s C 1 , X(') i s absolutely continuous, and g t ( t , * ) e x i s t s f o r almost a l l t i n [a,b], i t follows that x(t) = g f c(t,X(t)) + g (t,X(t))X(t) t e [a,b] a.e., and hence, L*(t,X(t) ,X(t)) < L(t , x ( t ) ,x(t)) - — <h(t) ,g(t,X(t))> t e [a,b] a By in t e g r a t i n g t h i s l a s t i n e q u a l i t y between a and b we obtain <h(b) ,g(b,X(b) )> - <h (a) ,g (a,X (a)) > + b L ( t , x ( t ) ,x(t) )dt , L * ( t , X ( t ) , X ( t ) ) d t which implies that f o r any arc x such that |x(t) - x ( t ) | < y for a l l t e [a,b] , and f o r X(t) = F ( t , x ( t ) ) we have: (3.19) J*(X) < J(x) The convexity of H(t,x,») and condition ( i i ) imply that H(t,x(t) ,p(t)) =<p(t),x(t)> - L(t , x ( t ) ,x(t)) a.e. By using t h i s equation and equation (3.18) i n (3.12) we conclude that 57 <h(b),g(b,X(b))> - <h(a),g(a,X(a))> + L * ( t , X(t), X(t))dt b L ( t , x ( t ) ,x(t) )dt , and hence, equations (3.9) and (3.15) imply that (3.20) J*(X) = J(x) . Since J*f(X) < J* (X) f o r a l l arcs X such that |x(t) -X(t) for a l l t e [a,b] , equations (3.19) and (3.20) imply that J(x) < J(x) for a l l x s a t i s f y i n g |x(t) - x ( t ) | < y f o r a l l t e [a,b] . Q.E.D. 3.4 Proof of Theorem (5) The proof i s based on using the r e s u l t s of the l a s t two sections. As we s h a l l see shortly, the extended Jacobi condition allows us to f i n d a subclass of the c l a s s of canonical transformations developed i n section (3.3). Any transformation of t h i s subclass takes the o r i g i n a l Hamiltonian H to a new Hamiltonian H* which s a t i s f i e s i n addition to the properties presented i n the previous section, the l o c a l concavity c r i t e r i o n given by Proposition (3.2). Moreover, such a transformation takes v i a (3.15) the function £ to the function £* which, by condition (d), s a t i s f i e s condition (3) of Proposition 58 (3.1). By applying t h i s proposition we obtain that the transformed arc X i s a strong l o c a l minimum for (P*). So, by using part (c) of Proposition (3.3) the proof w i l l be completed. Let Q(') be the L i p s c h i t z matrix function s a t i s f y i n g conditions (c) and (d) of Theorem (5). Lemma 3.1: There e x i s t a p o s i t i v e number a and two functions o F and P ; F°: { ( t , x ) : t e [a,b], | x - x ( t ) | < a } —>- TRn P: [a,b] -y ]R , such that P i s , F° s a t i s f i e s condition (A) of section (3.3), and (i) F° (*,x) i s continuous f o r a l l i = l,2,***,n , i ( i i ) F°(t,x(t)) = I , and (3.21) <F (t,x),P(t)> = - - <x - x(t),Q(t) (x - x(t))> + <P(t),x> Proof: Define the functions P and F = F l •o to be r \ -1 O P(t) - e. and 59 F°(t,x) = j <x-x(t),Q(t) (x - x(t))> + x x F°(t,x) = x. i = 2,•••,n . 1 1 Let q^(t) be the f i r s t column of the matrix Q(t) = (q 1 ( t ) ^ n ^ ) ) and define the function d(t,x) = <x - x ( t ) , q (t)> + 1 Since d(*,«) i s continuous and d ( t , x ( t ) ) = 1 , we can f i n d a p o s i t i v e number such that d(t,x) =(= 0 for a l l (t,x) e { ( t , x ) : t e [a,b] , l x - x ( t ) | < a } ; we denote the set of a l l o these (t,x) by S . e a r l i e r . gradient Take S as being the domain of the function F defined I t i s c l e a r that P i s C 1 , and F°(t,») i s C 2 with F (t,x) = I + x (x-x(t)rQ(t) o 0 o • 0 Then, F (t,x(t)) = I , F (*,x) i s continuous f o r l = l,"»n , and X X X . ' ' 1 det F (t,x) = d(t,x) . x 60 Hence, F (t,*) i s i n v e r t i b l e i n i t s domain S = {x: (t,x) e s}. with inverse g (t,*) . Since H (*,z) i s continuous, condition (b) of Theorem (5) z implies that z = (x,p) i s . Consequently, F°(*,x) and F°(',x) are L i p s c h i t z , and F^(t,x) = <x-x(t),Q(t)(x-x(t))> - <x(t),Q(t)(x-x(t))> 0 Thus, F t(t,«) e x i s t s whenever Q(t) e x i s t s (almost everywhere), and there e x i s t s a constant g such that, f o r t e [a,b] a.e., |F~(t,x)| < C l e a r l y our functions P and F s a t i s f y equation (3.21). Q.E.D. Now, l e t F° and P be the functions obtained from Lemma (3.1), and define (3.22) li (t) = p(t) - P(t) ; h° i s a C^-function on [a,b] . Consider the canonical transformation whose generating function has the form (3.8), where we take h(t) = li (t) 61 and F(t,x) = F°(t,x) . This transformation takes the arcs x, p and the Hamiltonian H(t,x,p) to the arcs X, P and to the Hamiltonian H*(t,X,P) defined by (3.5), (3.16), and (3.13) r e s p e c t i v e l y . The hypotheses of Theorem (5) allow us to apply Proposition (3.3). Hence, J(x) i s well-defined (possibly +°°) and J(x) i s f i n i t e . Moreover, to prove that x i s a strong l o c a l minimum f o r (P), by the same proposition, i t s u f f i c e s to show that the transformed arc X i s a strong l o c a l minimum f o r the problem (P*). To prove the optimality of X we w i l l apply Proposition (3.1), where Z* and H* are given by (3.15) and (3.13) r e s p e c t i v e l y . F i r s t , observe that the transformed Hamiltonian H* i s convex i n P . Also, by Proposition (3.3), we have that hypotheses (H*) and (H|) hold, and that Z = (X,P) s a t i s f i e s the Hamiltonian i n c l u s i o n s . Thus, the proof w i l l be complete i f we show that Z* s a t i s f i e s condition (3) of Proposition (3.1) arid that H*(t,*,P(t)) i s l o c a l l y concave around X . By Lemma (3.1), we have that F°(t,x(t)) = I and hence, equation (3.7) implies that g°(t,X(t)) = I . Thus, from (3.16) i t X follows that P(t) = p(t) - h°(t) . By using the d e f i n i t i o n of h° given by (3.22), we deduce that 62 (3.23) P(t) = P(t) = p(t) - h°(t) . Now, l e t a be the p o s i t i v e number given i n condition (d) of Theorem (5). The c o n t i n u i t y of g°(t,«) y i e l d s that there e x i s t s a p o s i t i v e number a* such that f o r any c* and d* i n 3Rn s a t i s f y i n g |c*|< a* , |d*| < a* , we can f i n d c and d i n 3Rn .with |c| < a , |d| < a such that (3.24) g°(a,X(a) + c*) = x(a) + c, F°(a,x(a) +c) = X ( a ) + c * , g°(b,X(b) +d*) = x(b) +d, F°(b,x(b) + d) = X(b) + d* Let c* and d* be such that |c*| < a* and |d*| < a* , and l e t c and d be the corresponding vectors s a t i s f y i n g |c| < a , |d| < a and equations (3.24). By using equations (3.15), (3.5), (3.24), (3.23) and condition (d) of Theorem (5), we obtain that £*(x(a) +c*,X(b) +d*) - £ * ( x ( a ) , X ( b ) ) = <h°(b) ,g°(b,X(b) +d*) -g°(b,X(b))> - <h°(a) ,g°(a,X(a) +c*) -g°(a,X(a))> + £(g°(a,X(a) + c*) ,g°(b,X(b) +d*)) - £(g°(a,X(a)) ,g°(b,X(b))) > <h°(b),d> - <h°(a),c> + <p(a),c> - <p(b),d> + j <d,Q(b)d> - j <c,Q(a)c> = <P(a),c> - <P(b),d> + j <d,Q(b)d> - j <c,Q(a)c> . 63 But, by using equations (3.21), (3.23), (3.24) and (3.5) we deduce that <P(a),c> - <P(b),d> + j <d,Q(b)d> - ^ <c,Q(a)c> = <P(a),F°(a,x(a) +c) - F°(a,x(a))> - <P(b) ,F°(b,x(b) +d) - F°(b,x(b))> = <P(a),c*> - <P(b),d*> . Thus, f o r any c* and d* s a t i f y i n g |c*| < a* and |d*| < a* we have: £ * ( X ( a ) +c*,X(b) +d*) - I* (X (a) ,X(b)) > <P(a),c*> - <P(b),d*> , which i s condition (3) of Proposition (3.1). I t remains to show that the Hamiltonian H* , defined by (3.13) and (3.11), i s l o c a l l y concave around X . For that we w i l l apply Proposition (3.2). From (3.9) and (3.10), we can express the o r i g i n a l v a r i a b l e s (x,p) i n terms of the new va r i a b l e s (X,P) as follows x = g°(t ,x) and (3.25) P = ( g ° ( t , x ) ) T -1 P + h°(t) 64 Let e and 6 be the p o s i t i v e numbers given i n hypothesis (H ), and l e t a^ia^^e) be the p o s i t i v e number obtained by Lemma (3.1). o o The c o n t i n u i t y of the functions g and g implies that there X e x i s t p o s i t i v e numbers a* and 6 * such that the following holds: f o r any t i n [a,b], and for any (X,P) s a t i s f y i n g | x - X ( t ) | < a* , | p - P ( t ) | < y* , the p a i r (x,p) , obtained by (3.9) and (3.25), s a t i s f i e s | x - x ( t ) j < a Q and | p - p ( t ) | < 6 . From equations (3.11), (3.9) and (3.25), the function H* (t, • ,P (t)) , defined on {X: | x - X ( t ) | < a*} , can be expressed as follows: (3.26) H*(t,X,P(t)) = H(t,g°(t,X), (g°(t,X)) T P (t) + h° (t) ) O X + <h°(t) ,g°(t,X)> - <(g°(t,X)) - 1g°(t,X) ,P(t)> . D i f f e r e n t i a t e equation (3.21) with respect to t , and then use equations (3.23) and (3.9) to get: <F°(t,g 0(t>;x)) ,P(t)> + <X,P(t)> = - y <g°(t,X) - x ( t ) ,Q(t) (g°(t,X) - x ( t ) ) > + <g°(t,X) - x(t) ,Q(t)x(t)> + <g°(t,X) ,P(t)> . By using (3.6), (3.7) and (3.23) i n the above equality, we obtain 65 (3.27) - <(g°(t,X)) 1 g ° ( t , X ) , P ( t ) > + <h°(t),g°(t,X)> = - j <g°(t,X) - x ( t ) ,Q(t) (g°(t,X) - x ( t ) ) > + <g°(t,X) - x(t) ,Q(t)x(t)> + <p(t) ,g°(t,X)> - <X,P(t)> . By d i f f e r e n t i a t i n g (3.21) with respect to x , and by using (3.9) and (3.7), we obtain that (g°(t,X)) T P(t) = - Q(t)(g°(t,X) - x ( t ) ) + P(t) . A Thus, from (3.25) and (3.23) i t follows that (3.28) p = - Q(t) (g°(t,X) - x ( t ) ) + p(t) . Therefore, by s u b s t i t u t i n g (3.27) and (3.28) i n (3.26), the expression of H* reduces t o : o (3.29) H*(t,X,P(t)) =H(t,g°(t,X) ,p(t) -Q(t) (g°(t,X) - x ( t ) ) ) - -j <g°(t,X) -x{t) ,Q(t) (g°(t,X) - x(t) )> + <g°(t,X) - x(t),Q(t)x(t)> + <p(t),g°(t,X)> - <X,P(t)> , where X e {X: | x - X ( t ) | < a * } . The c o n t i n u i t y of H {• ,z) and condition (b) of Theorem (5) imply that x and p are C"*" . Also, from (3.5) and (3.16) i t follows 1 that X and P and C , too. Since the function Q(*) i s L i p s c h i t z , 66 equation (3.29) implies that H (t,*,«) exists whenever Q(t) exists (almost everywhere). Thus, the transformed Hamiltonian at P = P(t) reduces to: (3.30) H*(t,X,P(t)) = 4 H Q(t,X,P(t)) whenever Q(t) exists 0 elsewhere From hypothesis (H^), we know that H(t,«,«) i s on { (x,p) : |x-x(t)| < e, |p-p(t)| < 6} . Also, we have that g°(t,-) 2 * is C . Then, the function H (t,«,P(t)), defined by (3.29), i s C 1 + and hence, H*(t,-,P(t)) i s C 1 + on {X: |x-X(t)|<a*} . To apply Proposition (3.2) we need to show that, for some * Y * > 0 , the matrix H (t,X,P(t)) i s negative semi-definite whenever XX * H exists on {(t,X,P(t)) : t e [a,b] , |x-X(t)| < Y*> Let t e [a,b] be such that Q(t) does not exist. From (3.30) i t follows that H*(t,X,P(t)) = 0 and hence, H* (t,X,P(t)) = 0 . XX Now, let t e [a,b] be such that Q(t) exists. By (3.30) we have that H*(t,X,P(t)) = H*(t,X,P(t)) . Hence, from (3.29) we can o * ic calculate the expressions of H and H , whenever the latter o o X XX exists, as follows: 67 (3.31) H* (t,X,P(t)) = (H (t,g°(t,X),p(t) -Q(t) (g°(t,X)-x(t)))+p(t))g°(t,X) o x A X - H (t,g°(t,X),p(t)-Q(t)(g°(t,X)-x(t)))Q(t)g°(t,X) p X + ( ( x ( t ) ) T Q ( t ) - (g°(t,X)-x(t))TQ(t))g°(t,X) - P ( t ) , and (3.32) H (t,X,P(t)) XX ((g°(t,X)) T,(-Q(t)g°(t,X)) T) D H (t,g°(t,X),p(t)-Q(t)(g°(t,X)-x(t))) z z g°(t,x) - Q ( t ) g x ( t , x ) + (D (g° (t,X)) ) (H (t,g°(t,X) ,p(t) -Q(t) (g (t,X)-x(t)) + p (t)) X X X - (D v(g°(t,X)) T) (Q(t)H (t,g°(t,X) , P ( t ) -Q(t) (g°(t,X) - x ( t ) ) ) X X Jp - (g°(t,X)) TQ(t)g°(t,X) + (D x(g°(t,X)) T) (Q(t)x(t) -Q(t) (g° (t, X)-x (t)) ) , where, f o r any matrix function A(t,X) any for and vector v , (D A(t,X))v = ( T — A(t,X)v — — A(t,X)v) . A O A , O A 1 n To complete the proof we need to e s t a b l i s h the following r e s u l t . Lemma 3.2 : If condition (c) of Theorem (5) holds, then there e x i s t s a p o s i t i v e number 3 such that f o r t e [a,b] and a l l x, p, E, W, and S s a t i s f y i n g 68 | x - x ( t ) | < 3, | p - p ( t ) | < 3 , | E - 11 < g , |w + Q ( t ) | < 3, |s + Q(t)| < 3 (whenever Q e x i s t s ) , t h e f o l l o w i n g h o l d s : (E T,W T) D H (t,x,p) z z w + S < 0 whenever D H (t,x,p) e x i s t s , z z P r o o f : We w i l l p r o ve t h e r e s u l t by c o n t r a d i c t i o n . Suppose t h a t t h e r e s u l t i s f a l s e . Then, f o r e v e r y p o s i t i v e i n t e g e r K such t h a t n n • K > max(e ,6) we can f i n d (t ,x ,p ) e [a,b] x T R X ] R where Q (t ) K K K K and D H (t ,x ,p ) e x i s t , r e IR n , and n x n - m a t r i c e s E , W , S Z Z K K K K K K K s a t i s f y i n g |WK + Q ( t K ) | < \ , |S K + Q(t K)| < | , | r R | = 1 , and (3.33) r K • [ ( E T , W T ) D z H z ( t K , X K , p K ) K W K + S K ] r K > ° S i n c e x, p and Q a r e c o n t i n u o u s , and s i n c e t h e i n t e r v a l [a,b] and t h e u n i t sphere o f ] R N a r e compact, we have ( a f t e r p a s s i n g t o a subsequence i f n e c e s s a r y ) t h a t 69 t K — t Q e [a,b], x K - x ( t o ) , p K - p ( t o ) , where |r| = 1 . Given that the function Q(•) i s L i p s c h i t z on the compact set [a,b] then, by [8, Section 3], the map t —y 3Q(t) i s upper semicontinuous on [a,b], where 3Q i s the generalized gradient of Q Hence, there e x i s t s a p o s i t i v e constant such that 3Q(t)| < M f o r a l l t e [a,b] On the other hand, for each t i n [a,b] we have Q(t ) e 3Q(t ) , so the sequence {Q(t )} converges to some matrix n . By using the K O upper semicontinuity of the map t —»• 3Q(t) we conclude that n e 9Q(t Q) , and hence S K _ _ % E _ 3 Q ( T O , . From hypothesis (H 2) we have that the map (t,z) y 3 H (t,z) z z i s upper semicontinuous. So, we can f i n d a p o s i t i v e constant M2 such that 3 H ( t , z ) | < M f o r (t„z) e N(e,6) . 70 Thus, {D H (t ,x ,p )} converges to some matrix v , and v i s Z Z K K K O O i n the set 3 H (t ,x(t ),p(t )) . We write v = z z o o o o a <5 • o o P> Y o o At the l i m i t the i n e q u a l i t y (3.33) becomes: r . [ (I, - Q(t ))v o o -Q(t o) - n Q ] r > 0 , which i s the same as r . [n - Q ( t ) Y Q(t ) +Q(t )B + 6 Q(t ) - a ] r < 0 , o o o o o o o o o where a 6 o o i Y o 'o e 3 H (t ,x(t ) ,p(t )) , and n e 3Q(t ) z z o o o o x o This i n e q u a l i t y contradicts condition (c) of Theorem (5) which i s assumed to be s a t i s f i e d . Q.E.D. Equations (3.9) and (3.28) express the o r i g i n a l v a r i a b l e s (x,p) i n terms of the new var i a b l e X when the second new var i a b l e ~ o P i s P(t) . By cont i n u i t y of g we can f i n d a p o s i t i v e number B* such that: whenever | x - X ( t ) | < B* , we have | x - x ( t ) | <B and | p - p ( t ) | < B , where x and p are given by (3.9) and (3.28), and where B i s the constant i n Lemma (3.2). Thus, f o r X e {X: | X - X ( t ) | < B*} , the function H* (t,X,P(t)) given by (3.32) can be written as: °XX 71 (3.34) H (t,X,P(t)) = XX ((E(t,X)) T,(W(t,X)) T)D H (t,x,p) z z E(t,X) W(t,X) + S(t,X) , where (3.35) E(t,X) = g x(t,X) , (3.36) (3.37) W(t,X) = - Q(t)g (t,X) A S(t,X) = D (g°(t,X)) T(H (t,x,p) + p(t)) X X X > - (D Cg°(t,X)) T) (Q(t)H (t,x,p)) P - (g°(t,X)) TQ(t)g°(t,X) + (D x(g°(t,X)) T) (Q(t)x(t) -Q(t) ( x - x ( t ) ) ) , and where x and p are given by (3.9) and (3.28), that i s , x = g°(t,X), p = - Q(t)(g°(t,X) - x(t)) + p(t) . From Lemma (3.1) we have that F (t,x(t)) = I , then, (3.7) implies that g°(t,X(t)) = I . Thus, (3.35) and (3.36) y i e l d that (3.38) E(t,X(t)) = I , W(t,X(t)) = - Q(t) By using condition (b) of Theorem (5), equation (3.37) implies that (3.39) S(t,X(t)) = - Q(t) . 7 2 Since E(•,•) and W(•,•) are continuous, we can f i n d a p o s i t i v e number Y ^ ( Y £ - 6*) such that f o r a l l t i n [a,b] and for a l l X s a t i s f y i n g | x - X ( t ) | < y£ we have | E ( t , X ) - l | < B , and jW(t,X) + Q(t) | < B , where 3 i s the constant i n Lemma ( 3 . 2 ) . On the other hand, the function S(t,*) i s defined whenever Q e x i s t s ( t e [a,b] a.e.). But,' i n the expression of S(*,*) a l l the terms other than Q are continuous i n (t,X) . Since Q(t) i s bounded on [a,b] , i t follows ic ie ic from ( 3 . 3 7 ) that there e x i s t s a p o s i t i v e number Y 2 ^ Y 2 ~ ® ^ such that for a l l t where Q(t) e x i s t s , and for a l l X s a t i s f y i n g | x - X ( t ) | we have |s(t,X) - S ( t , X ( t ) ) | = |s(t,X) + Q(t)| < B . Let y* = m i n i Y ^ j Y ^ ' Y * - 3* • Then, for a l l t such that Q(t) e x i s t s , and f o r a l l X s a t i s f y i n g | x - X ( t ) | < Y * we have: | x - x ( t ) | < B, | p - p ( t ) | < B, | E ( t , X ) - l | < B , |w(t,X) +Q(t) I < B , and IS(t,X) + Q(t)| < B -By using Lemma ( 3 . 2 ) and equation ( 3 . 3 4 ) , we conclude that H* (t,X,P(t)) < 0 , °XX 73 f o r a l l t such that Q(t) e x i s t s , and for a l l X e {X: | X - X ( t ) | < y*}. Therefore, H* x(t,X,P(t)) < 0 , whenever H (t ,X,P (t)) e x i s t s on the set { ( t , X , P ( t ) ) : t.e [a,b] , |X-X(t) <y*}. Q.E.D. 74 CHAPTER IV A Hamilton-Jacobi Approach In the c l a s s i c a l calculus of v a r i a t i o n s , the Hamilton-Jacobi i n e q u a l i t y was used to produce a s u f f i c i e n c y theorem. Also, t h i s approach was developed f o r optimal c o n t r o l problems [2], [14], [25], [26] and [27] . A s u f f i c i e n c y theorem was developed i n [16, Chapter' 2] f o r the generalized problem of Bolza (P) where one boundary value i s f i x e d ; x(a) = A . This r e s u l t introduced a generalized version of the Hamilton-Jacobi equation. In t h i s chapter we present a s u f f i c i e n c y theorem for the generalized problem of Bolza. One c o r o l l a r y of t h i s theorem i s a s u f f i c i e n c y c r i t e r i o n i n v o l v i n g the generalized Hamilton-Jacobi i n e q u a l i t y as opposed to equation. Thus, t h i s c r i t e r i o n i s an extension of a known c r i t e r i o n i n optimal c o n t r o l (involving the Hamilton-Jacobi inequality) to the generalized problem of Bolza. Another c o r o l l a r y of our theorem i s a s u f f i c i e n c y c r i t e r i o n which we employ i n the next chapter to unify the s u f f i c i e n c y theorem presented here and the one i n v o l v i n g the Jacobi condition; Theorem (5). Theorem 6: Assume that L i s L x8-measurable and that x i s a given admissible arc for (P) such that J(x) i s f i n i t e . Assume also that there e x i s t s a l o c a l l y L i p s c h i t z function W(t,x) defined f o r some y > 0 on { (t,x) : t e [a,b] , | x - x ( t ) | <y) and s a t i s f y i n g : (i) there e x i s t s a p o s i t i v e number p; p < y , such that f o r a l l c,d s a t i s f y i n g |c| < p and |d| < p we have 75 (4.1) W(a,x(a)+c) -W(a,x(a)) + W(b,x(b)) -W(b,x(b)+d) < £(x(a) + c,x(b) +d) - £(x(a) ,x(b)) , ( i i ) i f we define Z(t,x) = max{a+H(t,x,6): (a,g) e3W(t,x)} , we have: (4.2) Z(t,x) < Z(t,x(t)) f o r t e [a,b] a.e. , and d x (4.3) Z(t,x(t)) = --W(t,x(r.) ) - -L(t,x(t) ,x(t)) a.e. at Then J(x) i s well-defined (possibly +«=) for x near x , and x i s a strong l o c a l minimum f o r (P). Proof: Let x be any arc s a t i s f y i n g |x(t) - x ( t ) | < y f o r a l l t i n [a,b]. We want to show that J(x) i s well-defined. F i r s t , observe that W(*,') i s L i p s c h i t z and x i s absolutely continuous. Thus, —-W(t,x(t)) e x i s t s f o r almost a l l t i n [a,b] . at Define d • G = {t: — W ( t , x ( t ) ) exists} n {t: x(t) exists} ,-at G has a Lebesgue measure (b-a). For a point t i n G define f (T) = W(t + x, x(t) + xx(t)) , t 76 for T i n [0,e] , where e i s some p o s i t i v e number. By using the chain rule i n [8, Section 13], we obtain 3f t(0) c 3w(t,x(t)) • ( l , x ( t ) ) . On the other hand, from the f a c t that t i s i n G i t follows that x( t + x) = x(t) + rx(t) + O ( T ) . Hence, W(t + T , x ( t + t ) ) - W(t,x(t)) = W(t + x,x(t) + xx(t)) - W(t,x(t)) + 0(x) . Thus, f o r any t i n G we have: d f t ( T ) - f (0) -—W(t,x(t)) = l i m e 3f (0) c 3w(t ,x (t)) • (1 ,x (t)) Therefore, for a l l arcs x s a t i s f y i n g |x(t) - x ( t ) | < y i we get (4.4) — W ( t , x ( t ) ) e 3W(t,x(t))•(l,x(t)) a.e. dt From the d e f i n i t i o n of H we can rewrite Z as follows: Z(t,x(t)) = max sup{a + 3 .v - L (t ,x (t) , v) : v e TR1 , (a,3) e 3W(t,x(t))} . 77 Hence, by using (4.2) - (4.4) we obtain dt W(t,x(t)) - L ( t , x ( t ) , x ( t ) ) < Z(t,x(t)) ~ d t W ( t ' x ( t ) ) " L(t , x ( t ) ,x(t)) a.e. This implies that (4.5) L(t,x(t) ,x(t)) > — W ( t , x ( t ) ) - — W ( t , x ( t ) ) - L(t , x ( t ) ,x(t)) a.e. at at Since J(x) i s f i n i t e and L i s Lx B-measurable, (4.5) y i e l d s that J(x) i s well-defined (possibly +°°) f o r x near x . Now, l e t x be an arc s a t i s f y i n g |x(t) - x ( t ) | < p f o r a l l t i n [a,b] , where p i s the constant given i n (i) . By using (4.5) and (4.1) we deduce that J(x) - J(x) = £(x(a),x(b)) - £(x(a),x(b)) b b + L ( t , x ( t ) , x ( t ) ) d t - L ( t , x ( t ) , x ( t ) ) d t > W(a,x(a)) - W(a,x(a)) + W(b,x(b)) b - W(b,x(b)) + dt { W(t,x(t)) - W(t,x(t)) }dt = 0 . Thus, x provides a strong l o c a l minimum for (P). Q.E.D. The following r e s u l t provides a s u f f i c i e n c y c r i t e r i o n i n v o l v i n g the generalized Hamilton-Jacobi i n e q u a l i t y . I t i s a s p e c i a l case of Theorem (6); the function W i s assumed to s a t i s f y Z(t,x(t)) =0 , too. As we s h a l l show i n the next chapter, t h i s c r i t e r i o n i s i n f a c t equivalent to Theorem (6), when the Hamiltonian H(•,•,•) i s continuous. C o r o l l a r y 1: Assume that L i s Lx B-measurable and that x i s a given admissible arc f o r (P). Assume also that there e x i s t s a l o c a l l y L i p s c h i t z function W(t,x) defined f o r some y > 0 on { (t,x) : t £ [a,b] , | x - x ( t ) | < y} and s a t i s f y i n g : (a) there e x i s t s p > 0 ; p < y , such that f o r . a l l c , d s a t i s f y i n g |c| < p and fdj < p we have W(a,x(a) +c) - W(a,x(a)) + W(b,x(b)) - W(b,x(b) +d) < £(x(a) +c,x(b) +d) - I (x (a) ,x (b)) , (b) i f we define Z(t,x) = max{a +H(t,x,B): (a,B) e 3W(t,x)} , the generalized Hamilton-Jacobi i n e q u a l i t y holds; (4.6) Z(t,x) < 0 for t £ [a,b] a.e., 79 and (4.7) — W(t,x(t)) - L ( t , x ( t ) , x ( t ) ) = 0 a.e. at Then J(x) i s well-defined (possibly +°°) f o r x near x , J(x) i s f i n i t e , and x i s a strong l o c a l minimum for (P). Proof: From (4.7) i t follows that b » L ( t , x ( t ) , x ( t ) ) d t = W(b,x(b)) - W(a,x(a)) , a and hence, the a d m i s s i b i l i t y of x implies that J(x) i s f i n i t e . Now, from the d e f i n i t i o n of H , and from (4.4), (4.6) and (4.7) we obtain: Z(t,x(t)) = W(t,x(t)) - L( t , x ( t ) ,x(t)) = 0 a.e. dt Thus, a l l the conditions of Theorem (6) are s a t i s f i e d and hence, by t h i s theorem the proof i s completed. Q.E.D. The following c o r o l l a r y provides a s u f f i c i e n c y c r i t e r i o n f o r the generalized problem of Bolza. In the next chapter, t h i s r e s u l t w i l l serve two purposes. F i r s t , i t w i l l e s t a b l i s h the connection between Theorems (5) and (6). Second, i t w i l l provide another proof for Theorem (5). 80 1 Co r o l l a r y 2: Assume that L i s Lx B-measurable and that x i s C and admissible f o r (P) with J(x) f i n i t e . Assume also that there e x i s t a C^-function p(t) , a C^"-matrix function Q(t), and a p o s i t i v e number p s a t i s f y i n g : (1) L ( t , x ( t ) , x ( t ) +v) - L ( t , x ( t ) ,x(t)) ><p(t),v> fo r a l l v i n IR n , and for almost a l l t i n [a,b] , (2) f o r a l l c,d s a t i s f y i n g |c| < p and |d| < p £(x(a) + c,x(b) +d) - £(x(a) ,x(b)) ><p(a),c> 1 1 - <p(b),d> - — <c,Q(a)c> + — <d,Q(b)d> , (3) for a l l t i n [a,b] , and for a l l x; | x - x ( t ) j < p H(t,x,p(t) -Q(t) ( x - x ( t ) ) ) - H(t,x(t),p(t)) < - <p(t) ,x -x(t)> + j <x - x(t) ,Q(t) (x - x(t) )> - <x(t),Q(t) ( x - x ( t ) ) > . Then J(x) i s well-defined (possibly +°°) f o r x near x , and x i s a strong l o c a l minimum f o r (P) . Proof: Define (4.8) W°(t,x)=--j<x-x(t) , Q(t) ( x - x ( t ) ) > + <p(t),x> Since x, p are C"*" and Q i s L i p s c h i t z , equation (4.8) implies 81 t h a t W°(.,.) i s C 1 and W°(t,») i s C°° L e t c,d be such t h a t |c| < p and |d| < p . From c o n d i t i o n (2) and e q u a t i o n (4.8) i t f o l l o w s t h a t W°(a,x(a)+c) - W°(a,x(a)) +W°(b,x(b)) - W° (b,x (b) + d) = ~ \ < c ' Q ( a ) c > + <p(a),c> + |- <d,Q(b)d> - <p(b) ,d> < £(x(a) + c,x ( b ) +d) - £(x(a),x(b)) whence, c o n d i t i o n ( i ) o f Theorem (6) h o l d s . To p r o v e t h a t c o n d i t i o n ( i i ) i s s a t i s f i e d , we o b s e r v e t h a t Z ° ( t , x ) = max{a+H ( t ,x ,B) : (a,g) e 9 W ° ( t , x ) } W°(t,x) + H(t,x,W°(t,x)) By u s i n g (4.8) we o b t a i n : (4.9) Z°(t,x) = H ( t , x , p ( t ) - Q ( t ) ( x - x ( t ) ) ) - j <x - x ( t ) ,Q(t) (x - x ( t ) )> + <x ( t ) ,Q(t) ( x - x ( t ) )> + <p(t),x> From c o n d i t i o n (2) i t f o l l o w s t h a t Z°(t,x) , g i v e n by ( 4 . 9 ) , s a t i s f i e s r e l a t i o n (4.2) f o r a l l t i n [a,b] . On t h e o t h e r hand, t h e d e f i n i t i o n o f t h e H a m i l t o n i a n H and c o n d i t i o n (1) i m p l y t h a t 82 (4.10) H(t,x(t) ,p(t)) = <p(t),x(t)> - L(t,x(t) ,x(t)) a.e. Also, from (4.8) we have (4.11) ^ W°(t,x(t)) = <p(t),x(t)> + <p(t),x(t)> . Thus, equations (4.9) - (4.11) y i e l d that Z°(t,x(t)) = H(t,x(t) ,p(t)) + <p(t),x(t)> = <p(t),x(t)> + <p(t),x(t)> - L(t,x(t) ,x(t)) a.e. d o — W (t,x(t)) - L(t, x ( t ) ,x(t)) a.e. Therefore, a l l the conditions of Theorem (6) are s a t i s f i e d . Hence, by the same theorem the r e s u l t i s obtained. Q.E.D. Remark: If the conditions i n Theorem (6) and C o r o l l a r i e s (1) and (2) n ^ hold f o r a l l x i n 3R (not only i n a neighbourhood of x) , then i t n ~ follows that J(x) i s well-defined f o r a l l x i n 3R and that x i s a global minimum f o r (P) 83 CHAPTER V I n t e r r e l a t i o n s h i p s Between the S u f f i c i e n t Conditions The objective of t h i s chapter i s to e s t a b l i s h the connection between the d i f f e r e n t s u f f i c i e n t conditions developed i n the previous chapters f o r the generalized problem of Bolza. We have four approaches which are the "point convexity" (Theorem 3), the "Hamiltonian-conjugacy" (Theorem 5), the "modified Hamilton-Jacobi i n e q u a l i t y " (Theorem 6), and a new one given by Corollary (2) of Theorem (6). The r e l a t i o n s h i p s between these approaches are given i n figure (1). Figure (1) (Generalized H-J) Th.6 Cor.1 Th.6 Cor.2 (new) ( c l a s s i c a l conj.) Th.2 ->- Th. 3 (Pt. convexity) Th. 4 (convexity) 84 In figure (1) the arrow between two statements A and B , A B , indicates that under some conditions statement B i s a c o r o l l a r y of statement A . The l e t t e r G under the arrow means that A ->• B with-out conditions. D e f i n i t i o n : Let E be a subset of TRU having Lebesgue measure zero. Let f be a l o c a l l y L i p s c h i t z function from TR1 to TR . The set 9 f(x) i s defined to be the convex h u l l of elements of the form E m = lim Vf(x.) , i->oo where {x^} converges to x , Vf(x^) e x i s t s and x^ £ E f o r each i . The following r e s u l t shows that, i n the smooth case, Theorem (6) i s equivalent to the c r i t e r i o n i n v o l v i n g the generalized Hamilton-Jacobi i n e q u a l i t y , that i s , Corollary (1) of Theorem (6). Proposition 5.1; Let x be an admissible arc f o r (P). If the Hamiltonian H(t,x,p) i s continuous with respect to (t,x,p), then the hypotheses of Theorem (6) and C o r o l l a r y (1) are equivalent. Proof: Assume that the hypotheses of Theorem (6) are s a t i s f i e d f o r some L i p s c h i t z function W . Since for each t i n [a,b] the generalized gradient 9W(t,x(t)) i s compact, 9W(t,x(t)) i s uniformly bounded on [a,b] . Thus, the c o n t i n u i t y of H implies that the function Z(« ,x(«)) , given by Z(t,x(t)) = max{ct + H(t,x(t) ,B) : (a,3) e 9W(t,x(t))} , 85 i s uniformly bounded on [a,b] . Define (5.1) W(t,x) = W(t,x) Z(s,x(s))dt on {(t,x) : t e [a,b] , | x - x ( t ) | < y} , where y i s the constant given i n Theorem (6). The function W(*,«) i s l o c a l l y L i p s c h i t z . From condition (i) of Theorem (6) i t follows that f o r |c| < p and |d| < p we have: W(a,x(a) +c) - W(a,x(a)) + W(b,x(b)) - W(b,x(b) +d) = W(a,x(a) +c) - W(a,x(a)) + W(b,x(b)) - W(b,x(b) + d) Z(s,x(s))ds + Z(s,x(s))ds < £(x(a) +c,x(b) +d) - £(x(a) ,x(b)) . Thus, W s a t i s f i e s condition (a) of Coro l l a r y (1). Since the function W s a t i s f i e s condition ( i i ) of Theorem t (6) and r e l a t i o n (4.4), and since the i n t e g r a l J Z(s,x(s))ds i s a L i p s c h i t z , then the set A , defined by A = {t: equation (4.3) holds} n {t: — W (t ,x (t) ) exists} n {t: dt Z(s,x(s))ds e x i s t s and t i s a Lebesgue point of Z(s,x(s))} , 86 has a Lebesgue measure (b-a) . From the d e f i n i t i o n of W and from equation (4.3) i t follows that cl "~ -T— W(t,x(t)) e x i s t s f o r a l l t i n A , and dt - | l W(t,x(t)) = W(t,x(t)) - Z(t,x(t)) dt dt = L(t , x ( t ) ,x(t) ) . Hence, W s a t i s f i e s equation (4.7). Now, define (5.2) Z(t,x) = max{a+H(t,x,B): (a,3) e.9W(t,x)} . I t remains to show that Z s a t i s f i e s i n e q u a l i t y (4.6). We w i l l prove t h i s by c o n t r a d i c t i o n . Define B = {t: Z(t,x) < Z(t,x(t)) f o r a l l x e {x: | x - x ( t ) | < y}} . Since W s a t i s f i e s (4.2), the set B has a Lebesgue measure (b-a). Suppose that Z does not s a t i s f y (4.6). Then, there e x i s t (t ,x ) o o and a p o s i t i v e number e such that t e B , x - x ( t ) < y , and o 1 o o ' (5.3) Z(t ,x ) > e . o o Define 87 t S = { ( t , x ) : t e B, ^-a t Z ( s , x ( s ) ) d s e x i s t s , t i s a a Lebesgue p o i n t o f Z ( s , x ( s ) ) , and VW(t,x) e x i s t s } , and l e t E be t h e complement o f S ; E = { ( t , x ) : ( t , x ) / S} . The s e t E has measure z e r o . Hence, from [3, P r o p o s i t i o n (1.11)] i t f o l l o w s t h a t 9W(t ,x ) = 9 W(t ,x ) , o o E o o (where, f o r a g i v e n f u n c t i o n f , t h e s e t 9 f ( x ) was d e f i n e d e a r l i e r ) E Thus, u s i n g (5.3) and ( 5 . 2 ) , we deduce t h a t t h e r e e x i s t s (a ,3 ) e 9„W(t ,x ) such t h a t o o E o o (5.4) a + H ( t ,x ,3 ) > E . o o o o S i n c e (a ,3 ) e 9 W(t ,x ) , t h e n (a ,3 ) i s o f t h e form o o E o o o o r r 1 2 (a ,3 ) = y = I x .a ,r.) , O O . L . 1 1 . L . 1 1 1 1=1 1=1 where 88 (5.5) I. = l i m VW(t..,x..) , 1 ll in j->-oo •> -> the sequence (t..,x..) converges to (t ,x ) , (t..,x..) I E and ID l ] o o ' i ] i ] y VW(t,.,x..) e x i s t s f o r each j . iD ID The convexity of H(t,x,«) implies that r 1 2 a + H(t x B ) < I \.U.7 + H(t ,x ,1)) , O O O O . ^ . , 1 1 O O l 1=1 and hence, from (5.4) we deduce that f o r some k e {l,2,«««,r} we have: l} + H(t ,x ,£?) > 0 . k o o k Since H(•,*,•) i s continuous, the l a s t i n e q u a l i t y and equation (5.5) imply that f o r j large we get W V + H ( t k j ' x k j ' w x ( t k j ' x k j ) ) > 0 ' where (t . ,x, .) e S and VW(t, . ,x, .) e x i s t s f o r each j kD kD kD kD J Thus, from equation (5.1) i t follows that, f o r j large W V + H ( t k j ' x k j ' w x ( t k j ' x k j , ) > ^ V ^ V 5 But, we have that t . e B . Then, kD V W + H ( t k j ' X k j ' W x ( t k j ' x k j ) ) > Z ( t k j ' V 89 The l a s t i n e q u a l i t y contradicts the d e f i n i t i o n of Z . Therefore, Z s a t i s f i e s (4.6) and the proof i s completed. Q.E.D. Remark: The co n t i n u i t y assumption on H i n Proposition (5.1) cannot be removed. This i s shown i n the following example. Example: Consider the generalized problem of Bolza: 1 (P) minimize J(x) = ¥^ o }(x(0)) + ^ 0 j ( x ( l ) ) + 1 ( x 2 + ^ - + l ) d t , 1/2 " 4 0 where (•) denotes the i n d i c a t o r function of the set A . A In t h i s problem we have: 1 2 v 2 L ( t , x , v ) = ~YJ2 ( x + T " + 1 ) and £ ( x r x 2 } = * { o } ( x i ) + ^ o } ^ • I t i s c l e a r that L i s L x B-measurable. Consider the arc x(t) = 0 f o r t e [0,1] . So, 90 J(x) = £(x(0) ,x(D) + L ( t , x ( t ) , x ( t ) ) d t dt .1/2 = 2 , 0 0 hence J(x) i s f i n i t e and x i s admissible f or (P). The Hamiltonian corresponding to our problem (P) i s : H(t,x,p) = sup{<p,v> - L(t,x,v) : v e TR.} 1 2 v 2 = sup{pv ^ x + y + 1) : v £ IRj ,.1/2 2 1 . 2 . , Thus, H(0,x,p) = - oo and hence, the Hamiltonian i s not continuous. Define W(t,x) = 0 . Then, Z(t,x) = max{a+H(t,x,6): (a,B) e 3W(t,x)} 1 ,2 (x + 1) . .1/2 C l e a r l y , f o r a l l t e [a,b] and x e IR we have: Z(t,x) = - 1 (x 2 + 1) < - = Z(t,x(t)) , .1/2 .1/2 and 91 Z(t,x(t)) = - i — = ^ - W ( t , x ( t ) ) - L(t , x ( t ) ,x(t) ) at Thus, a l l the conditions of Theorem (6) are s a t i s f i e d and hence, x = 0 i s a strong l o c a l minimum for (P) . Now, suppose that there e x i s t s a l o c a l l y L i p s c h i t z function W s a t i s f y i n g conditions (a) and (b) of Coro l l a r y (1). In p a r t i c u l a r , W must s a t i s f y equation (4.7), that i s , — W(t,0) = L(t,0,0) = -r-pr a.e. dt \/2 whence, W (t,0) = -^-pr for t e [0,1] a.e. t X/2 Thus, W(',0) i s not L i p s c h i t z . But t h i s conclusion contradicts the L i p s c h i t z property of W . Therefore, there e x i s t s no L i p s c h i t z function W s a t i s f y i n g the hypotheses of Corollary (1) . However, W(t,x) = 0 s a t i s f i e s the conditions of Theorem (6). The following r e s u l t shows that, under some assumptions, the s u f f i c i e n c y theorem inv o l v i n g the Jacobi condition (Theorem (5)) i s a s p e c i a l case of the s u f f i c i e n c y c r i t e r i a obtained i n the previous chapter (Theorem (6) and C o r o l l a r i e s (1) and (2)). Proposition 5.2: Assume that a l l the hypotheses of Theorem (5) hold, and that the matrix function Q s a t i s f y i n g the extended Jacobi 92 condition i s C"*" . Then a l l the hypotheses of Coro l l a r y (2) , and hence of Theorem (6), are s a t i s f i e d . Remark: I f the conditions of Proposition (5.2) hold, and i f the function H(«,x,p) i s continuous on [a,b], then Propositions (5.1) and (5.2) imply that the hypotheses of Corollary (1) are s a t i s f i e d . Proof of Proposition (5.2): From hypothesis (H^) we have that L i s L x8-measurable. Let x, p be the arcs given by Theorem (5). Since H (•,•,•) z i s continuous, condition (b) of Theorem (5) implies that z = (x,p) i s C*~ . Also, the conclusion of the same theorem gives that J(x) i s f i n i t e . Let Q be the C''"-matrix function s a t i s f y i n g conditions (c) and (d) of Theorem (5). Our proof w i l l be completed when we show that Q s a t i s f i e s condition (3) of Corol l a r y (2). Define (5.6) Z(t,x) = H(t,x,p(t) -Q(t) ( x - x ( t ) ) ) +<p(t),x> - j <x - x(t) ,Q(t) (x - x(t) )> + <x(t),Q(t) ( x - x ( t ) ) > . Since H(t,.,.) i s C^~+ near z = (x,p) , there e x i s t s a p o s i t i v e number a such that f o r a l l t i n [a,b] and f o r a l l x i n {X: | x - x ( t ) | < a} the function Z(t,«) i s C^"+ . From (5.6) we can c a l c u l a t e Z and 93 and Z whenever the l a t t e r e x i s t s ; xx (5.7) Z (t,x) = H ( t , x , p ( t ) - Q ( t ) ( x - x ( t ) ) ) x and - H -(t,x,p(t) -Q(t) ( x - x ( t ) ) ) Q ( t ) + p(t) - ( x - x ( t ) ) T Q ( t ) + ( x ( t ) ) T Q ( t ) , (5.8) Z x x ( t , x ) = H x x ( t ' x ' P ( t ) ~ 2 ( t ) ( x - x ( t ) ) ) - H (t,x,p(t) - Q(t) (x - x(t)) )Q(t) - Q(t)H (t,x,p(t) - Q(t) ( x - x ( t ) )) + Q(t) H (t,x,p(t) -Q(t) (x - x(t) ) )Q(t) " Q(t) (I, -Q(t))D H ( t , X , p ( t ) - Q ( t ) ( x - x ( t ) ) ) Z Li I ' -Q(t) - Q(t) From (5.7) and condition (b) of Theorem (5) i t follows that (5.9) Z (t,x(t)) = 0 x Thus, to prove that condition (3) of Corollary (2) i s s a t i s f i e d i t s u f f i c e s to show that f o r a l l t i n [a,b] the function Z ( t , - ) i s concave near x . But, using Lemma (3.2) and equation (5.8), we deduce that there e x i s t s a p o s i t i v e number 3 such that f o r a l l 94 t e [a,b] and f o r a l l x i n {x: | x - x ( t ) | < (3} we have: Z (t,x) ^  0 whenever i t e x i s t s , xx Hence, from Proposition (3.2) we conclude that f o r a l l t i n [a,b] the function Z(t,») i s concave near x . Q.E.D. Remark: Proposition (5.2) suggests a new proof f o r Theorem (5), where the extended Jacobi condition i s s a t i s f i e d by a C^-function Q . However, as we s h a l l now see, the function Z defined by (5.6) plays the same ro l e i n the proof of Proposition (5.2) that the transformed * Hamiltonian H q plays i n the proof of Theorem (5) (section 3.4). To see that, we notice that equations (5.8), (3.29), and (3.9) imply that (5.10) H*(t,F°(t,x),P(t)) = Z(t,x) - <F°(t,x),P(t)> , o where F° and P = p are obtained from Lemma (3.1). From (5.10) i t follows that, whenever Z e x i s t s , xx (5.11) (F°(t,x)) T H* (t,F°(t,x),P(t))F°(t,x) = X o X XX Z (t,x) - (D (F°(t,x)) T)(H* (t,F°(t,x) ,P(t)) + P(t)) . X X X X o A From (3.31) i t follows that H* (t,X(t),P(t)) = - P(t) . Also, from Lemma (3.1) we know that the matrix F (t,x) i s i n v e r t i b l e x for x near x , and that F°(t,x(t)) = I . Thus, by using equation (5.8), Lemma (3.2), and Proposition (3.2), equation (5.11) implies that H (t,«,P(t)) i s concave near X(t) = F (t,x(t)) . o '-1 0 Moreover, i f we take P = 6 1 = , as i n the proof of Lemma (3.1), then, equation (5.10) becomes: H (t,F (t,x),P(t)) = Z(t,x) . o Hence, i n t h i s case, the two functions H and Z coincide. o The r e l a t i o n between the "point convexity" c r i t e r i o n (Theorem 3) and those of the previous chapter i s given by the following p r o p o s i t i o n . Proposition 5.3: I f the hypotheses of Theorem (3) are s a t i s f i e d with 1 p being C , then the hypotheses of Theorem (6) are also s a t i s f i e d . If i n addition x i s C^ " , the hypotheses of Corollary (2) also hold. Remark: Assume that the Hamiltonian H(t,x,p) i s continuous. I f the hypotheses of Theorem (3) hold for a ^ - f u n c t i o n p , Propositions (5.1) and (5.3) y i e l d that the hypotheses of Corollary (1) also hold. Proof of Proposition (5.3): Suppose that the hypotheses of Theorem (3) are s a t i s f i e d . Then L i s L xB-measurable and, by Theorem (3), J(x) i s f i n i t e . 96 I f both arcs x and p are C 1 , the functions Q(t) = 0, x, and p s a t i s f y the conditions of Coro l l a r y (2). 1 Now, assume that only p i s C Let y be the p o s i t i v e number given i n Theorem (3). Define (5.12) W(t,x) = <p(t),x> , where (t,x) e { (t,x) : t e [a,b] , | x - x ( t ) | < y] . Then W(«,*) i s c 1 . Consider c and d i n IRn such that |c| < y and |d| < y . From (5.12) and condition ( i i i ) of Theorem (3), i t follows that W(a,x(a)+c) - W(a,x(a)) +W(b,x(b)) -W(b,x(b)+d) = <p(a),c> - <p(b),d> < lU(a) +c,x(b) +d) - £(x(a) ,x(b)) . Hence, condition (i) of Theorem (6) i s s a t i s f i e d . Let Z(t,x) = max{a + H(t,x,B) : (a,B)e 3"W(t,x)} , where W i s given by (5.12). Thus, (5.13) Z(t,x) = <p(t),x> + H(t,x,p(t)) . Using condition ( i i ) of Theorem (3), equation (5.13) implies that 97 Z(t,x) - Z(t,x(t)) = <p(t) ,x - x(t)> + H(t,x,p(t)) - H(t,x(t) ,p(t)) < 0 . Thus the function Z(t,x) , defined by (5.13), s a t i s f i e s r e l a t i o n (4.2). To complete the proof i t remains to show that equation (4.3) holds. From the d e f i n i t i o n of the Hamiltonian H , and from condition (i) of Theorem (3), i t follows that H(t,x(t),p(t)) = <p(t),x(t)> - L ( t , x ( t ) , x ( t ) ) a.e. Also, equation (5.12) implies ^ W ( t , x ( t ) ) = <p(t),X'(t)> + <p(t),x(t)> a.e. Thus, equation (5.13) y i e l d s Z(t,x(t)) = <p(t),x(t)> + H(t,x(t),p(t)) • • • = <p(t),x(t)> + <p(t),x(t)> - L( t , x ( t ) ,x(t)) a.e. = - W { t , x ( t ) ) - L ( t , i ( t ) ,x(t)) a.e. at Therefore equation (4.3) i s s a t i s f i e d . Q.E.D. The following r e s u l t w i l l be needed l a t e r i n t h i s chapter. 98 Lemma 5.1: Let A ( t ) , B(t) and C(t) be n x n-continuous matrix functions on [a,b], where B(t) and C(t) are symmetric. Let Q b be a symmetric constant matrix. Assume that there e x i s t s a C^-matrix function Q q such that Q 0 ( t ) i s symmetric and solves, for a l l t i n [a,b], Q(t) - Q(t)B(t)Q(t) + A T ( t ) Q ( t ) + Q(t)A(t) + C(t) > 0 Q(b) = Q b • Then there e x i s t s a C^-matrix function Q such that Q(t) i s symmetric and s a t i s f i e s f o r a l l t i n [a,b] Q(t) - Q(t)B(t)Q(t) + A T ( t ) Q ( t ) + Q(t)A(t) + C(t) > 0 Q(b) < Q b . Proof: Define the matrix function (5.14) F(t) = Q (t) - Q (t)B(t)Q (t) + A T ( t ) Q (t) + Q (t)A(t) + C ( t ) . o o o o o Then F(t) > 0 for a l l t i n [a,b] . Now define the f i r s t order parametric matrix d i f f e r e n t i a l equation: (5.15) Q(t) = Q(t)B(t)Q(t) - A T ( t ) Q ( t ) - Q(t)A(t) - C(t)> + F(t) + X (5.16) Q(b) = Q b - X , 9y where F(t) i s defined by (5.14), Q(t) i s symmetric, and the matrix A i s of the form X = 11 0 nn The hypotheses of t h i s lemma imply that f o r A = 0 the matrix function QQ(t) solves the d i f f e r e n t i a l equation (5.15) and condition (5.16). 2 Write equations (5.15) and (5.16) as an n -vector d i f f e r e n t i a l equation with a given boundary value at b . Since the matrices B ( t ) , C ( t ) , Q Q ( t ) , X , and Qfa are symmetric, 2 h - n equations are redundant. Hence, equations (5.15) and (5.16) can be 2 considered as being — - — f i r s t order parametric d i f f e r e n t i a l equations with given boundary values at b . Apply Theorem (4.1) of the appendix of [13] to our 2 n + n parametric d i f f e r e n t i a l equations, then convert back to the matrix form. Doing so, i t r e s u l t s that there e x i s t p o s i t i v e numbers X , A , and a symmetric continuous matrix function Q(t), such 11 nn x # that Q(t) and A = 1 1 ° 0 nn s a t i s f y equations (5.15) and (5.16). Since the functions A(*), B(*), C(*), and Q (•) are continuous, equation (5.15) implies that Q(') i s C"^  . On the other hand, we have that 100 F(t) > 0 , and X > 0 . Thus, from equations (5.15) and (5.16) i t follows that the matrix function Q s a t i s f i e s Q(t) - Q(t)B(t)Q(t) + A T ( t ) Q ( t ) + Q(t)A(t) + C(t) and = F(t) + X > 0 Q(b) = Q b - X < Q b . Q.E.D. Now consider the generalized problem of Bolza (P) with a given boundary value x(a) = A . Then the function Z.(x ,x ) takes the form (5.17) £ ( x 1 # x 2 ) = , i ' { A } ( x 1 ) + £°(x 2) , where £° i s a function from TRn to IR u {+°°}. , and (x) is the i n d i c a t o r function of the set {A} , that i s , i f x = A Lf x =(  A 101 For the case when £ i s of the form (5.17), Lemma (5.1) suggests a method for the v e r i f i c a t i o n of the extended Jacobi condition and condition (d) of Theorem ( 5 ) . This method i s given by the following r e s u l t . C o r o l l a r y 3: Consider the generalized problem of Bolza (P), where £ i s given by (5.17). Let z = (x,p) be a given arc with x admissible f o r (P) . Assume that £°(*) i s C 2 near x(b) with p(b) = - £ (x(b)) , H(t,») i s C near z , and the Jacobian D H (*,z)' i s continuous. Also assume that there e x i s t s a C^-matrix z z function Q q such that, f o r a l l t i n [a,b] , QQ^) i s symmetric and solves the Rica t t i - m a t r i x d i f f e r e n t i a l equation Q(t) - Q(t)H (t,x(t) ,p(t) )Q(t) + Q(t)H (t,x(t) ,p (t) ) x x PP px + H ( t , x ( t ) , p ( t ) ) Q ( t ) - H ( t , x ( t ) , p ( t ) ) = 0 , xp X X with Q(b) = £° (x(b)) . XX Then, there e x i s t s a C 1-matrix function Q such that, f o r a l l t i n [a,b], Q(t) i s symmetric and s a t i s f i e s conditions (c) and (d) of Theorem ( 5 ) . Proof: The f a c t that H(t,«) and £°(') are continuous allows us to define the following q u a n t i t i e s : 102 A(t) = H ( t , x ( t ) ,p(t) ) px B(t) = H (t,x(t) ,p(t)) PP C(t) = - H (t,x(t) ,p(t)) XX Q h = 1° (x(b)) . £> xx Since D H (*,z) i s continuous, so are the matrix functions A ( t ) , z z B ( t ) , and C(.t) . Hence, by Lemma (5.1), we can f i n d a C^-matrix function Q such that, for a l l t i n [a,b] , Q(t) i s symmetric and s a t i s f i e s and Q(t) - Q(t)H ( t , x ( t ) ,p(t))Q(t) + Q(t)H ( t , x ( t ) ,p(t)) + H ( t , x ( t ) , p ( t ) ) Q ( t ) - H x x ( t , x ( t ) , p ( t ) ) > 0 , Q(b) < 1° (x(b)) XX Thus, condition (c) of Theorem (5) holds. It remains to show that the function Q s a t i s f i e s condition (d) of Theorem (5). Let ^ ] _ ' * * * ' ^ n k e the eigenvalues of the p o s i t i v e d e f i n i t e matrix (x(b)) - Q(b); these eigenvalues are p o s i t i v e . Let X = min{X^: i = l , , , , , , n } . Then, for a l l d i n 3Rn , (5.18) <d,(£° x(x(b)) - Q(b))d> > x|d|2 . Since t°(•) i s C 2 and p(b) = - £°(x(b)) , we have for x x near x(b) 103 (5.19) £°(x) = £°(x(b)) - <p(b) ,x-x(b)> + ^-<x-x(b),£° (x(b)) (x-x(b))> 2 xx + |x - x(b) o(x - x(b)) , where l i m 0 ( x - x ( b ) ) = Q x->x(b) Ix - x(b) | Choose a > 0 such that Y+irf-^ ° f o r |d| < a . Let c , d be vectors i n 3Rn such that | d | < a . If c =j= 0 , equation (5.17) implies that £(x(a) +c,x(b) +d) = + 0° , and hence condition (d) of Theorem (5) i s s a t i s f i e d . I f c = 0 and |d| < a , then (5.17) - (5.19) imply that £(x(a) ,x(b) +d) - £(x(a) ,x(b) ) = £°(x(b) +d) - £°(x(b)) = - <p(b),d> + y <d,Q(b)d> + \ <d,(£° x(x(b)) - Q ( b ) ) d > + |d| 0(d) > - <p(b),d> + j <d,Q(b)d> I I2 X o(d), > - <p(b),d> + j <d,Q(b)d> . 104 Thus, condition (d) of Theorem (5) holds. Q.E.D. Remark: For the case when both boundary values are f i x e d (x(a) = A and x(b) = B) , a r e s u l t s i m i l a r to C o r o l l a r y (3) can be stated. In t h i s case, the condition p(b) =-- £°(x(b)) and the boundary value Q(b) = £° (x(b)) are removed, xx The following r e s u l t shows that under c e r t a i n hypotheses the "modified Hamilton-Jacobi" approach (Theorem 6) i s a c o r o l l a r y of the "Hamiltonian conjugacy" approach (Theorem 5). Hence, t h i s r e s u l t gives a c r i t e r i o n for the v e r i f i c a t i o n of the extended Jacobi condition and condition (d) of Theorem (5). Proposition 5.4: Consider the generalized problem of Bolza (P) with a f i x e d boundary value x(a) = A . Assume that the hypotheses of Theorem (6) are s a t i s f i e d by a L i p s c h i t z function W . Suppose that 3 (i) W(t,«) i s C , the functions W(*,x), W (•••,x) , 1 2 and W (*,x) are C , W, (t,«) i s C , and xx t W (t,x(t)) = W . ( t , x ( t ) ) , W. (t,x(t)) =W (t,x(t)) , tx xt txx xxt ~ 2 ( i i ) f o r p(t) = W x(t,x(t)), the function H(t,«) i s C near z = (x,p) and the functions H (*,z) and D H (•• ,z) are z z z continuous on [a,b] , ( i i i ) x(t) = H ( t , x ( t ) , p ( t ) ) for t e [a,b] . Then the hypotheses of Theorem (5) are s a t i s f i e d f o r some C^-function Q 105 Remark: In the case where the hypotheses of Coro l l a r y (2) hold, the conditions of Theorem (6) and condition (i) of Proposition (5.4) are automatically s a t i s f i e d . Proof of Proposition (5.4): Let x be the admissible arc given i n Theorem (6), and l e t p(t) = W x(t,x(t)) . The hypotheses of our proposi t i o n imply that hypotheses (H^) and (H^) are s a t i s f i e d . Condition ( i i i ) and the d e f i n i t i o n of p imply that x and ,1 Since W{«,«) i s C 1 , i t follows that p are C (5.20) Z(t,x) = max{a+H(t,x,3): (a,g) e 3W(t,x)} = W. (t,x) + H(t,x,W (t,x)) . t x Hence, f o r a l l t e [a,b] Z(t,x(t) ) = W (t,x(t) ) + sup{<p(t) ,v> - L(t,x( t ) , v ) : v e TRn} . But equation (4.3) and the smoothness of x imply that Z(t,x(t)) = w t ( t , x ( t ) ) + <p(t),x(t)> - L ( t , x ( t ) , x ( t ) ) f o r t € [a,b]. From these l a s t two e q u a l i t i e s we deduce that f o r a l l v e JR1 and fo r a l l t e [a,b] , L(t, x ( t ) ,x(t) + v) - L(t,x(t) ,x(t)) ><p(t),v> . 106 Therefore condition (a) of Theorem (5) i s s a t i s f i e d . Now, r e l a t i o n (4.2) of Theorem (6) means that f o r a l l t e [a,b] and f o r x near x , max Z(t,x) = Z(t,x(t)) . 2 Equation (5.20) implies that Z(t,«) i s C , and hence f o r a l l t i n [a,b] (5.21) Z (t,x(t)) = 0 , and (5.22) Z (t,x(t)) < 0 xx Using (5.20), equation (5.21) implies that f o r a l l t e [a,b] , (5.23) W ( t , x ( t ) ) + H (t,x(t) ,p(t)) +H (t,x(t) ,p(t))W (t,x(t)) t X X jp X X = o . However, p(t) = W (t,x(t)) , and so p ( t ) = W . ( t , x ( t ) ) + x(t)W ( t , x ( t ) ) . Xt XX Thus, from condition ( i i i ) and equation (5.23) we have p(t) = - H ( t , x ( t ) , p ( t ) ) f o r a l l t e [a,b] , and hence condition (b) of Theorem (5) holds. From equation (5.20), equation (5.22) can be written as (5.24) Z (t,x(t)) = w. (t,x(t)) +H ( t , x ( t ) , p ( t ) ) xx txx X X + W (t,x(t))H (t,x(t) ,p(t)) xx px + H (t,x(t) ,p(t) )W (t,x(t) ) xp X X +W (t,x(t))H (t,x(t),p(t))W (t,x(t)) X X PP X X ' + W (t,x(t))H (t,x(t) ,p(t) ) <0 . xxx p Define (5.25) Q (t) = - W (t,x(t)) . O X X The function Q q i s symmetric and C"'" with d e r i v a t i v e Q (t) = - W (t,x(t)) - W ( t , x ( t ) ) x ( t ) . o xxt xxx Using equations (5.24) and (5.25), and condition ( i i i ) , i t follows that Q s a t i s f i e s *o Q (t) - Q (t)H (t,x(t) ,p(t) )Q (t) + H (t,x(t) ,p(t) )Q (t) o o PP o xp o + Q (t)H ( t , x ( t ) , p ( t ) ) - H ( t , x ( t ) , p ( t ) ) >0 , o px X X and Q (b) = - W (b,x(b)) O X X Hence, by Lemma (5.1) , we can f i n d a C 1-matrix 108 function Q such that Q(t) i s symmetric and s a t i s f i e s condition (c) of Theorem (5), and (5.26) Q(b) < - W (b,x(b)) . xx To complete the proof i t remains to show that the function Q s a t i s f i e s condition (d) of Theorem (5), where £ i s given by (5.17) . Let p be the p o s i t i v e number given by Theorem (6), and l e t c and d be vectors i n 3Rn such that |d| < p . Since x i s admissible f o r (P), x(a) = A . Hence, i f c ^ 0 we have that £(x(a) +c,x(b) +d) = + °° , and condition (d) of Theorem (5) i s s a t i s f i e d . On the other hand, i f . c = 0 and |d| < p , then r e l a t i o n (4.1) reduces t o : (5.27) £(x(a) ,x(b) + d) - £(x (a) ,x (b)) >W(b,x(b)) -W(b,x(b)+d) . However, r e l a t i o n (5.26) implies that f o r a l l d e 3Rn , <d, (-w .(b,x(b)) -Q(b))d> > x|d|2 , where X i s the smallest eigenvalue of -W^ x(b,x(b)) - Q(b) . Also, f o r d near zero we have - W(b,x(b) +d) = - W(b,x(b)) - <Wx (b,x (b)) ,d> - J-<d,W (b,x(b))d> - |d| o(d) , 2 xx 1 1 109 where lim = 0 d-K) I d I Choose a > 0 , a ^ p , such that f o r dI < a 2 i , i Thus f o r d < a we have: W(b,x(b)) - W(b,x(b) +d) = - <p(b),d> +y<d,Q(b)d> + - <d, (-W (b,x(b)) -Q(b))d> d| 0(d) > - <p(b),d> + - <d,Q(b)d> > - <p(b),d> + - <d,Q(b)d> Hence, using (5.27), we conclude that condition (d) of Theorem (5) i s s a t i s f i e d . Q.E.D. The "Hamiltonian conjugacy" approach (Theorem 5) i s a s u f f i c i e n c y c r i t e r i o n f o r the problem (P) where the Hamiltonian H i s not necessa r i l y concave-convex. However, the "point convexity" approach (Theorem 3) requires a c e r t a i n convexity assumption on the 110 Hamiltonian H . In the following r e s u l t , we show that under c e r t a i n assumptions the "point convexity" approach i s a s p e c i a l case of the "Hamiltonian conjugacy" approach. Proposition 5.5: Consider the generalized problem of Bolza (P) where t i s given by (5.17). Assume that the hypotheses of Theorem (3) are s a t i s f i e d by an arc z = (x,p) . Suppose that H(t,«) i s C 2 near z , H («,z) and D H (• ,z) are continuous, and z z z (-p(t) ,x(t)) = H (t,x(t) ,p(t)) for t e [a,b] Then the hypotheses of Theorem (5) are s a t i s f i e d f or some C^-function Q • Remark: In Proposition (5.5) we can assume the hypotheses of Theorem (4) instead of those of Theorem (3). In t h i s case the Hamiltonian equations hold automatically. Proof of Proposition (5.5): From the hypotheses of Theorem (3) we 2 have that hypothesis (H^ holds. Also, since H(t,«) i s C , and H (*,z) and D H (*,z) are continuous, hypothesis (H ) of Z Z Z 2. Theorem (5) i s s a t i s f i e d . Condition (i) of Theorem (3) i s the same as condition (a) of Theorem (5). Also, by assumption, we have that condition (b) of Theorem (5) i s s a t i s f i e d . Thus, i t remains to show that conditions (c) and (d) of Theorem (5) hold. Since H (*,z) i s continuous, the Hamiltonian equations 1 imply that x and p are C Condition ( i i ) of Theorem (3) means that f o r a l l t e [a,b] and for x near x , max{H(t,x,p(t)) + <p (t) ,x>} = H(t,x(t) ,p(t)) « + <p(t),x(t)> , which implies H ( t , x ( t ) , p ( t ) ) < 0 f o r a l l t e [a,b] xx Hence, f or a l l t i n [a,b] , the function Q Q(t) = 0 solves Q(t) - Q(t)H (t,x(t) ,p(t) )Q(t) + Q(t)H (t,x(t) ,p(t) ) + H (t,x(t) ,p(t) )Q(t) - H x x ( t , x ( t ) ,p(t)) > 0 with Q(b) = 0 Thus, by Lemma (5.1), we conclude that there e x i s t s a C^-matrix function Q s a t i s f y i n g condition (c) of Theorem (5) and Q(b) < 0 Let y k e the p o s i t i v e number given by condition ( i i i ) of Theorem (3), and l e t c and d be vectors i n TRU such that |d| < y • If c 7^  0 , then equation (5.17) and the a d m i s s i b i l i t y of 112 x imply that £(x(a) + c,x(b) +d) = + » . Hence, when c f 0 condition (d) of Theorem (5) i s s a t i s f i e d . Let c = 0 and |d| < y . Since Q(b) < 0 , condition ( i i i ) of Theorem (3) y i e l d s £(x(a) ,x(b) +d) - £(x(a) ,x(b) ) >-<p(b),d> > - <p(b),d> + Q(b) . Therefore condition (d) of Theorem (5) holds. Q.E.D. Consider the c l a s s i c a l calculus of v a r i a t i o n s problem (P^) minimize J^(x) = b L ( t , x ( t ) , x ( t ) ) d t , subject to x(a) = A , x(b) = B , where the minimum i s taken over a l l piecewise smooth functions x: [a,b] >• IRn . The problem (P^) can be written as a generalized problem of Bolza, where 113 (5.28) * ( V x 2 ) = f { A } ( X ; L ) + . { B ) ( x 2 ) (Recall that * c ( * ) i s the i n d i c a t o r function of the set C.) Thus, the "Hamiltonian conjugacy" c r i t e r i o n (Theorem 5) i s applicable to Our i n t e n t i o n i s to show that for the problem (P^ ) Theorem (5) i s a g e n e r a l i z a t i o n of the " c l a s s i c a l conjugacy" c r i t e r i o n (Theorem (2)). However, to see the connection between condition (4) of Theorem (2) and the extended Jacobi condition (condition (c) of Theorem (5)) we need the following r e s u l t . Lemma 5.2: Let A ( t ) , B ( t ) , and C(t) be nxn-continuous matrix functions on [a,b] such that, for a l l t i n [a,b], B(t) and C(t) are symmetric and B(t) i s p o s i t i v e d e f i n i t e . Then the following are equivalent: (i) there e x i s t s a C^-matrix function Q(') on [a,b] such that for a l l t i n [a,b], Q(t) i s symmetric and s a t i s f i e s Q(t) - Q(t)B'(t)Q(t) + A T ( t ) Q ( t ) + Q(t)A(t) + C(t) > 0 , ( i i ) there e x i s t s a C^-matrix function Q(*) on [a,b] such that for a l l t i n [a,b], Q(t) i s symmetric and s a t i s f i e s Q(t) - Q(t)B(t)Q(t) + A T ( t ) Q ( t ) + Q(t)A(t) + C(t) = 0 , ( i i i ) there e x i s t s no n o n t r i v i a l function h solving 114 (B 1 C t ) h ( t ) - B 1 ( t ) A ( t ) h ( t ) ) + A T ( t ) B 1(t)h(.t) at - C(t)h(t) - A T ( t ) B _ 1 ( t ) A ( t ) h ( t ) = 0 , with h(a) = h(c) = 0 for some c e (a,b] . Proof: I t s u f f i c e s to prove the following implications: ( i i i ) -> ( i i ) , ( i i ) ( i ) , and (i) ( i i i ) . Suppose that ( i i i ) holds. By [12, Theorem 10.2], i t follows that there e x i s t two matrix functions U (t) and V (t) such that, o o f o r a l l t i n [a,b], det U (t) 0 , o U T ( t ) V (t) = V T ( t ) U (t) , o o o o and the p a i r (U (t),V (t)) solves the matrix system o o J U(t) = A(t)U(t) + B(t)V(t) V(t) = C(t)U(t) - A T ( t ) V ( t ) Define Q (t) = - V (t)U 1 ( t ) o o o Then Q (•) i s , and Q (t) i s symmetric. Furthermore, Q o o o solves 115 Q(t) - Q(t)B(t)Q(t). + A T ( t ) Q ( t ) + Q(t)A(t) + C.(t) = 0 for a l l t e [a,b] . Thus condition ( i i ) i s s a t i s f i e d . The i m p l i c a t i o n ( i i ) -> (i) i s obtained by applying Lemma (5.1). Suppose that (i) holds f or some matrix function Q Q ^ ) • Define Z Q ( t ) = Q Q(t) - Q o ( t ) B ( t ) Q Q ( t ) + A T ( t ) Q Q ( t ) + Q (t)A(t) + C(t) Then for a l l t i n [a,b], Z (t) i s p o s i t i v e d e f i n i t e . Let U (t) o o be the s o l u t i o n of the matrix d i f f e r e n t i a l equation U(t) = (A(t) - B(t)Q (t) )U(t) , U(a) = I , where I i s the n x n - i d e n t i t y matrix, and l e t V (t) be o V Q ( t ) = - Q Q ( t ) U (t) for a l l t e [a,b] Then, for a l l t i n [a,b] we have det U (t) f 0 , o U T ( t ) V (t) = V T ( t ) U (t) , and the p a i r (U (t),V (t)) solves the o o o o o o matrix system U(t) = A(t)U(t) + B(t)V(t) V(t) = - (Z Q ( t ) - C ( t ) ) U ( t ) - A T ( t ) V ( t ) . 116 Thus by [12, Theorem 10.2] we conclude that there e x i s t s no n o n t r i v i a l s o l u t i o n h(t) on [a,b] sol v i n g (B 1 ( t ) h ( t ) -B 1 ( t ) A ( t ) h ( t ) ) + A T ( t ) B 1 ( t ) h ( t ) - C(t)h(t) at - A T ( t ) B _ : L ( t ) A ( t ) h ( t ) + Z (t)h(t) = 0 , o with h(a) = h(c) = 0 for some c e (a,b] . But t h i s i s equivalent, by [12, Theorem 10.3], to saying that f o r every i n t e r v a l [a,3] c [a,b] we have: (1) I(n;a,3) = { [B 1 ( t ) n ( t ) - B 1 ( t ) A ( t ) T 1 ( t ) ] n ( t ) + [-A T(t)B 1 ( t ) n ( t ) - (-c(t) - A T ( t ) B 1 ( t ) A ( t ) +z (t)n]-n}dt o > 0 fo r every absolutely continuous vector function n such that 2 n(a) = n(3) = 0 , and n i s of c l a s s L [a,3] , and (2) I(n;a,g) = 0 i f and only i f n = 0 . Using the f a c t that zQ(t) i s p o s i t i v e d e f i n i t e on [a,b], i t follows that f o r every i n t e r v a l [a,B] c [a,b] 117 I (n;a,B) = i(n;a,B) + Z (t)irndt o s a t i s f i e s conditions (1) and (2). Therefore, by [12, Theorem 10.3], condition ( i i i ) holds. Q.E.D. The r e l a t i o n s h i p between Theorem (2) and Theorem (5) i s given by the following p r o p o s i t i o n . Proposition 5.6: Consider the c l a s s i c a l calculus of v a r i a t i o n s problem (P^). Assume that f o r some function x the hypotheses of Theorem (2) hold. Then there e x i s t s a C^-matrix function Q such that x ( t ) , p(t) = L^ (t ,x (t) ,x (t) ) , and Q s a t i s f y the hypotheses of Theorem (5). Furthermore, condition (4) of Theorem (2) i s equivalent to condition (c) of Theorem (5), where Q i s C 1 . Remark: For the v e r i f i c a t i o n of the " c l a s s i c a l Jacobi condition" Proposition (5.6) suggests a c r i t e r i o n which, as we s h a l l see i n the example given l a t e r , can be easier to apply. Proof of Proposition (5 .6): Let x be the C^-function given by Theorem (2). Define (5.29) p(t) = L ( t , x ( t ) , x ( t ) ) f o r a l l t e [a,b] . 3 ~ 1 1 Since L(•,»,•) i s C and x i s C , then p i s C The Hamiltonian of the problem (P^) i s defined to be 118 H(t,x,p) = sup{<p,v> - L(t,x,v) : v e 3Rn } Consider the equation: (5.30) p = L v(t,x,v) From condition (3) of Theorem (2) we have that L (t) > 0 . Usinq w (5.29) and the i m p l i c i t function theorem f o r (5.30), we conclude 2 that there e x i s t a p o s i t i v e number e and a C -function v(t,x,p) on N(e,e) such that f o r a l l (t,x,p) e N(e,e) (5.31) p = L v(t,x,v(t,x,p)) and v ( t , x ( t ) , p ( t ) ) = x(t) Hence, condition (2) of Theorem (2) implies that f or a l l (t,x,p) eN(e (5.32) H(t,x,p)-= <p,v(t,x,p)> - L(t,x,v(t,x,p)) . 2 Thus H (•,•,•) i s C on N(e,e) , and hypotheses (H^) and (H 2) of Theorem (5) hold. Condition (2) of Theorem (2) and equation (5.29) imply that condition (a) of Theorem (5) i s s a t i s f i e d . By d i f f e r e n t i a t i n g equation (5.32) with respect to x and p r e s p e c t i v e l y and by using (5.31) we get: (5.33) H x(t,x,p)= - L^(t,x,v(t,x,p)) H^(t,x,p) = v(t,x,p) . 119 Hence condition (1) of Theorem (2) and equations (5.29) and (5.31) y i e l d that condition (b) of Theorem (5) holds. Since the function Z(x^,x^) has the form (5.28), condition (d) of Theorem (5) i s s a t i s f i e d f o r a l l c and d i n lR n . The proof w i l l be completed when we show that condition (4) of Theorem (2) i s equivalent to condition (c) of Theorem (5), where Q i s C 1 . F i r s t , l e t us write condition (4) of Theorem (2) i n terms of the Hamiltonian. By d i f f e r e n t i a t i n g equation (5.31) we obtain: (5.34H v (t,x,p) = - L 1 ( t , x , v ( t , x , p ) ) L (t,x,v(t,x,p) X vv vx v (t,x,p) = (t,x,v(t,x,p) ) . Now d i f f e r e n t i a t e equations (5.33), then substitute v and v x p from (5.34) to get H (t,x,p) = - L (t,x,v(t,x,p)) -?v.2s. X X + L (t,x,v(t,x,p))L 1 ( t , x , v ( t , x , p ) ) L (t,x,v(t,x,p)), X V vv vx H ( t , x , p ) = - L (t,x,v(t,x,p))L 1 ( t , x , v ( t , x , p ) ) , X 1-? X V w and H (t,x,p) = L 1 ( t , x , v ( t , x , p ) ) pp w 120 Hence, using the f a c t that x(t) = v ( t , x ( t ) , p ( t ) ) , we conclude that L (t,x(t) ,x(t)) = H 1 ( t , x ( t ) ,p(t)) w PP • _-i L (t,x(t) ,x(t) ) = - H (t,x(t) ,p(t) )H X (t,x(t) ,p(t) ) xv xp PP and L ( t , x ( t ) , x ( t ) ) = - H ( t , x ( t ) , p ( t ) ) X X X X + H ( t , x ( t ) , p ( t ) ) H - 1 ( t , x ( t ) , p ( t ) ) H (t,x(t) ,p(t)) . xp PP px Thus condition (4) of Theorem (2) can be written i n terms of the Hamiltonian as follows: there e x i s t s no n o n t r i v i a l function h solving (H 1(t)h(t) -H 1 (t)H (t)h(t)) + ft (t)H 1 ( t ) h ( t ) dt pp pp px px pp + ft (t)h(t) - H (t)H 1 ( t ) H (t)h(t) = 0 , xx xp pp px with h(a) = h(c) = 0 f o r some c i n (a,b] . (For a function cf> the notation $ (t) designates cj> (t ,x (t) ,p (t)) .) Therefore, i f we take A(t) = H (t) , B(t) = H (t) , and C(t) = - ft ( px pp xx Lemma (5.2) implies that condition (4) of Theorem (2) i s equivalent to condition (c) of Theorem (5), where i n the l a t t e r the function Q i s C 1 . Q.E.D. Remark: As we have just seen, i f conditions (1), (2) and (3) of Theorem 121 (2) hold, then conditions (a) and (b) of Theorem (5) are s a t i s f i e d . The equivalence between the c l a s s i c a l Jacobi condition and the extended Jacobi condition brings up a new point of view concerning the former condition i n the c l a s s i c a l calculus of v a r i a t i o n s . As we s h a l l see below, the c l a s s i c a l Jacobi condition can now be interpreted as being e s s e n t i a l l y a necessary and s u f f i c i e n t condition f o r the existence of a c e r t a i n kind of canonical transformation, f o r which the transformed Hamiltonian i s l o c a l l y concave-convex. Suppose that x and F(t,x) are given functions and a Q i s a p o s i t i v e r e a l number such that x.:' [a,b] > IRn F: { (t,x) : t e [a,b] , Ix - x ( t ) I < a } • TP? . We w i l l make the following assumptions: 3 (B) F(t,-) i s C and i n v e r t i b l e with inverse g(t,X) ; the functions F(*,x) , F (*,x) , and F (• ,x) ( i = 1, • • • ,n) are C*~ ; ^ x xx. F (t,«) i s C ; and F (t,x(t)) = I , F (t,x) = F (t,x) , X X t "CX F ,.(t,x) = F t (t,x) f o r a l l i = l , - - - , n xx.t txx. ' ' Coroll a r y 4: Consider the c l a s s i c a l calculus of v a r i a t i o n s problem 3 (P ) . Assume that !,(•,•,•) i s C and that conditions (1), (2) 122 and (3) of Theorem (2) are s a t i s f i e d by a C^-function x . Then condition (4) of Theorem (2) i s a necessary and s u f f i c i e n t condition f o r the existence of a canonical transformation of the form (3.8) (where F = g 1 s a t i s f i e s hypothesis (B)) , f o r which the transformed Hamiltonian H i s l o c a l l y concave-convex. Proof: Since conditions (1), (2), and (3) of Theorem (2) hold and 3 since L (•,•,•) i s C , the Hamiltonian H takes the form (5.32) and i t i s C 2 . Assume that condition (4) of Theorem (2) also holds. ' By Proposition (5.6), there e x i s t s a C^-matrix function Q such that x , p(t) = L^ (t ,x (t) ,x (t)) , and Q s a t i s f y the conditions of Theorem (5). Let F° and P be the functions obtained by Lemma (3.1). Since the function Q i s , the function F° can be chosen so that i t also s a t i s f i e s hypothesis (B). Define h° , as i n (3.22) , h° = P - P . Consider the canonical transformation whose generating function has the form (3.8), where h(t) = h°(t) F(t,x) = F°(t,x) . From (3.30) i t follows that the transformed Hamiltonian corresponding to t h i s transformation i s 123 H*(t,X,P) = H*(t,X,P) , where H* i s given by (3.29) . This new Hamiltonian, as we have seen i n section (3.4), i s convex i n P and l o c a l l y concave i n X around the transformed arc X . Conversely, suppose that there e x i s t a C^-function h°(t) and a function F° s a t i s f y i n g hypothesis (B), such that the transformed Hamiltonian H* , v i a the canonical transformation given by (3.8), i s l o c a l l y concave-convex. f Since the functions H(t,«,«) , g°(t,«) = F ^ ( t , * ) , and o 2 * g t ( t , ' ) are C , the transformed Hamiltonian H given by (3.13) i s equal to H* and hence H*(t,*,«) i s C 2 near the arcs X(t) = F°(t,x(t)) P(t) = p('t) - h°(t) . From the remark following Proposition (5.6), conditions (1), (2) and (3) of Theorem (2) imply that condition (b) of Theorem (5) holds. Hence x and p are C"*~ and so are X and P . Now, using the f a c t that H*(t, ,P(t)) i s concave and C 2 -we obtain (5.35) H * ( t , X ( t ) , P ( t ) ) < 0 for a l l t e [a,b] XX For any matrix function A(t,X) and for any vector v i n 3Rn we define the matrix (D A(t,X))v = ( ^ A(t,X)v g|-A(t,X)v) 1 n 124 I f we set (5.36) G(t,X) = (g°(t,X)) T -1 equation (3.11) y i e l d s and H*(t,X,P) = H x g ° + H pD x(GP) + h°g x - ( G P ) T g ° x - g°D x(GP) (5.37) H x x(t,X,P) ((g°) T,(D x(GP)) T)D zH z D (GP) X + (D v(D v(GP)) T) (H - g°) X X p t o T • o + (D x(g°) l) (H x + h°) " ( Dx ( G P ) ) TC- < X ) T D X ( G P ) " (Dx(<x)T)(GP) ' where the functions H , g° and h° and t h e i r d e r i v a t i v e s are evaluated r e s p e c t i v e l y at (t,g°(t,X),G(t,X)P+h°(t)) , (t,X) , and t . Define the C^-symmetric matrix function Q (t) = - D v(G(t,X)P(t)) O X x=x(t) 125 From hypothesis (B) i t follows that g°(t,X(t)) = I , A and hence equation (5.36) implies (5.38) Q o(t) = - V G ( t , X ) P ( t ) ) | x = X ( t ) Thus, = V ( g > , x ) ) T P ( t ) ) | x = x ( t ) (5.39) Q o(t) = D x(( g°(t,X)) TP(t)| x = x ( t ) + D x a g x t ( t ' x ) ) ^ ( t ) ) l x = x ( t ) + D x ( D x ( ( g ^ ( t , X ) ) T P ( t ) ) X ( t ) ) | x = x ( t ) By differentiating (5.36) with respect to X , we get Dx(G(t,X)P) =- (g°(t,X))T (D x(g°(t,X)) T)(g x(t,X)) T P and consequently, using g (t,X(t)) = I and equation (5.38), we obtain: 126 (5.40) D x (D x (G(t ,X)P(t ) ) ) T X(t) l x = x ( t ) = Q 0 ( t ) D x ( g ° ( t , X ) X ( t ) ) i x = x ( t ) - D x ( D x ( ( g ° ( t , x ) ) T P ( t ) ) x ( t ) | x = x ( t )  + D x ( ( 5 x ( t ' x ) ) T ^ t , > l x - x ( t ) 2 o ( t ) • We observe that H (t , i(t) ,p(t)) + h°(t) = - p(t) + h°(t) = - P(t) . /s ^ o ^ Moreover, since g(t,X(t)) = x(t) and g (t,X(t)) = I , i t follows that H (t,x(t) , P ( t ) ) - g t ( t , X ( t ) ) = x(t) - g t ( t , X ( t ) ) = X(t) , and g° t(t,X(t)) + D x(g°(t,X)X(t))J x = x ( t ) = g° ( t , X ( t ) ) + D x ( ( g x ( t , X ) ) T X ( t ) ) | x = x ( f c ) = 0 . Xt Hence, by using (5.36) - (5.40) i n (5.35) we conclude that f o r a l l t e [a,b] , 1 2 7 H* (t,X(t),P(t)) (I, -Q (t))D H (t,x(t) ,p(t)) o z z -QQ(t) + eo<t)(g^(tfx(t)) + V g x ( t ' x ) i ( t ) ^x-xcty ) + (g°x(t,x(t)) + Dx((g°(t,x))Tx(t))|x=.(t))Qo(t) - Q (t) *o = ( l , - Q ( t ) ) D H (t,x(t) ,p(t) ) o z z -Q (t) o - Q (t) < 0 . o Thus, by Lemma ( 5 . 1 ) , there e x i s t s a C^-matrix function Q such that Q(t) i s symmetric and s a t i s f i e s condition (c) of Theorem ( 5 ) . Therefore, by Proposition ( 5 . 6 ) , we conclude that condition (4) of Theorem ( 2 ) holds. Q.E.D. The following example shows that i n the c l a s s i c a l s e t t i n g condition (c) of Theorem ( 5 ) may be simpler to v e r i f y than condition (4) of Theorem ( 2 ) . Example: Consider the c l a s s i c a l v a r i a t i o n a l problem: 5TT (P^) minimize * 2 4x . ( 2 7 x T - J ) dt , ( t + - ) x(0) = X(5TT) = 0 . For t h i s problem we have 128 2 4x 2 L(t,xv) = 27 v - ^ ( t + J ) 3 Then L(',',') i s C on [0,5ir] x TR x JR , and - 8x L (t,x,v) = 54v , L (t,x,v) = v x 3 ( t + J ) L (t,x,v) = 54 , L (t,x,v) = L (t,x,v) = 0 , w xv vx and — 8 L (t,x,v) = x x 4 3 ( t + J ) Let x(t) = 0 for a l l t e [0,5ir] . C l e a r l y the function x i n C^ and admissible f o r (P^) . Further-more , we have (t,x(t) ,x(t) ) = L (t,x(t) ,x(t) ) = 0 , dt v x and L (t,x,v) = 54 (>0) vv Thus the function L(t,x,*) i s s t r i c t l y convex and conditions (1), (2) and (3) of Theorem (2) are s a t i s f i e d . 129 To prove the optimality of our candidate k = 0 for the problem (P^) we intend to use Theorem (2). So we need to show that condition (4) of Theorem (2) holds. The Jacobi equation (equation (1.7)) corresponding to t h i s problem i s : (5.41) h(t) + — 4 h ( t ^ 3 = 0 f o r a l l t e [0,5ir] . 2 7 ( t + J ) To check whether we have a conjugate point to a = 0 i n the i n t e r v a l (0,5TT]- we need to f i n d the zeros of the general so l u t i o n of (5.41). But equation (5.41) i s not e a s i l y solved, and hence we are unable to v e r i f y condition (4) of Theorem (2). However, i n t h i s s i t u a t i o n , the usual t o o l f or v e r i f y i n g condition (4) i s the comparison theorem [12]. We observe that f o r t e (0,5TT] , 4 JL_ 2 7 ( t + i ) 3 " 1 6 with equality at t = 0 . Consider now the second order d i f f e r e n t i a l equation: (5.42) h(t) + ^ r h ( t ) = 0 f o r t e [0,5TT] . 16 The general s o l u t i o n of (5.42) which vanishes when t = 0 i s : h(t) = A s i n — t e [0,5TT] 130 So, i t follows that h(4ir) = 0 . Thus the comparison theorem does not give any information about the zeros of the solutions of equation (5.41) . The Hamiltonian corresponding to (P^) i s : 2 A  2 H(t ,X,p) = . (t+-) 2 Then H(',',*) i s C , H(t,x,») i s convex, and H(t,*,p) i s not concave but rather convex. Let p(t) = L (t,x(t) ,x(t)) = 0 for t s [0,4ir] , and l e t Q o(t) = 4 2 (t + J ) Then Q (t) - Q (t)H (t)Q (t) + Q (t)H (t) + H (t)Q(t) o o pp o o px xp - f t (t) 1 0 2 5 8 xx lA_ 4.3 C A fJ_ 4.4 ,^  4.3 ( t + j ) 5 4 ( t + j ) ( t + J ) 108t + 119 „ ^ ^ _ . . • = j-r > 0 f o r t e [0,4TT] . 54 (t + j ) Thus condition (c) of Theorem (5) holds and hence Proposition (5.6) implies that condition (4) of Theorem (2) i s s a t i s f i e d . Thus x = 0 131 solves (P ) . 132 CHAPTER VI Application to Optimal Control Problems In t h i s chapter we develop s u f f i c i e n c y c r i t e r i a f o r optimality i n optimal c o n t r o l . Concerning t h i s question we can f i n d i n the l i t e r a t u r e two c r i t e r i a . The f i r s t uses the "convexity" approach [24] . I t requires the Hamiltonian to be concave i n x and convex i n p . The second s u f f i c i e n c y approach employs the Hamilton-Jacobi i n e q u a l i t y ([2], [14], [25], [26] and [27]). Thus, these c r i t e r i a are p a r a l l e l to the ones given by Theorems (4) and (6) for the generalized problem of Bolza. Our aim i s to apply the new approach (Corollary 2) and the "Hamiltonian Jacobi" approach (Theorem 5) to optimal c o n t r o l problems and obtain two new s u f f i c i e n c y c r i t e r i a there. Let f, g, and £° be given functions; ^ , n ni n f: IR x 3R y IR g: 1 x i s- 3R 1°: IR" • IR, and l e t U be a subset of IRm . The autonomous optimal c o n t r o l problem i s defined follows: b g(x(t) ,u(t) )dt (C) minimize J(x,u) = -c°(x(b)) + over a l l absolutely continuous functions (arcs) x: [a,b] -> IRn , and measurable functions u: [a,b] -> IRm s a t i s f y i n g 133 (6.1) x(t) = f (x(t) ,u(t)) a.e • r (6.2) x(a) = A (6.3) u(t) e U a.e. The Hamiltonian of the problem (C) i s : (6.4) H(x,p) = sup{<p,f(x,u)> - g(x,u): u e U} . n D e f i n i t i o n : Let x be an arc from [a,b] to IR and u.be a measurable function from [a,b] to lR m . The p a i r (x,u) i s admissible f o r (C) i f i t s a t i s f i e s (6.1) - (6.3). D e f i n i t i o n : An admissible p a i r (x,u) i s a strong l o c a l minimum for (C) i f there e x i s t s a p o s i t i v e r e a l number y such that (x,u) minimizes J(x,u) over a l l admissible p a i r s (x,u) s a t i s f y i n g Theorem 7: Let (x,u) be an admissible p a i r f o r (C) such that x i s C^ " . Assume that U i s compact, and the functions f (x,u) and g(x,u) are continuous f o r x near x and for u i n U . Suppose that there e x i s t a C^-function p(t) , a C^-matrix function Q(t) , and a p o s i t i v e number p s a t i s f y i n g : x(t) - x(t) | < y fo r a l l t e [a,b] (i) <p(t), f (x(t) ,u(t))> - g(x(t),u(t)) > <p(t), f(x(t),u)> - g(x(t),u) 134 f o r a l l u e U , ( i i ) £°(x(b)+d) - £°(x(b)) > - <p(b),d> + j <d,Q(b)d> for a l l d: |d| < p , ( i i i ) H(x,p(t) - Q(t) (x - x ( t ) ) ) - H(x(t) ,p(t)) < - <|(t) ,x - x(t)> + y <x - x(t) ,Q(t) (x - x(t))> - <x(t) ,Q(t) (x - x(t) )> , f o r a l l t e La,b] , and for a l l x: | x - x ( t ) | < p . Then (x,u) provides a strong l o c a l minimum f o r (C) . Proof: To prove t h i s r e s u l t we f i r s t convert the optimal c o n t r o l problem (C) to a generalized problem of Bolza (P ), then we apply Corollary (2) of Theorem (6) to (p ) . Define (6.5) £(x 1,x 2) = V { A } ( x 1 ) + £°(x 2) ., and (6.6) L(x,v) = inf{g(x,u): v = f ( x , u ) , u e U} . Consider the generalized problem of Bolza: b (P c) minimize J c ( x ) = £(x(a),x(b)) + L ( x ( t ) , x ( t ) ) d t 135 over a l l arcs x from [a,b] to 3Rn , where t and L are defined by (6.5) and (6.6) r e s p e c t i v e l y . Define the function J(x,p,q) = sup{<q,u> + <p,f(x,u)> - g(x,u): u £ U } . Since the set U i s compact and f and g are continuous for x near x , the function J(x,p,q) s a t i s f i e s the boundedness condition for x near x . On the other hand, condition (i) and equation (6.6) imply (6.7) L ( x ( t ) , x ( t ) ) = g(x(t),u(t)) a.e. , and hence J c ( x ) i s f i n i t e . Thus, by Theorem (1), we have that L i s measurable and that (x,u) i s a strong l o c a l minimum f o r (C) i f x i s a strong l o c a l minimum for ' F o r t* i e optimality of x we w i l l apply C o r o l l a r y (2). Condition ( i i ) implies that condition (2) of C o r o l l a r y (2) holds, where I i s given by (6.5). Also, from Section (1.1) we know that the problems (C) and (P^) have the same Hamiltonian. Thus, condition ( i i i ) i s the same as condition (3) of Corollary (2). It remains to show that condition (1) of C o r o l l a r y (2) holds. Let v be any vector i n TRn . Consider t e [a,b] such that (6.7) holds. Define A(t,v) = {u £ U: x(t) + v = f(x(t),u)} If the set A(t,v) i s empty, equation (6.6) implies that 136 L ( x ( t ) , x ( t ) +v) = °° , and hence L(x(t) ,x(t) + v) - L(x(t) ,x(t)) ><p(t),v> . If A(t,v) i s not empty, then the infimum i n (6.6) i s attained at some u s a t i s f y i n g u e U , and x(t) + v = f(x ( t ) , u ) . Hence, condition (i) y i e l d s L ( x ( t ) , x ( t ) +v) - L(x(t) ,x(t)) =g(x(t),u) - g (x (t) ,u (t)) > < p ( t ) , f ( x ( t ) , u ) - f ( x ( t ) , u ( t ) ) > = <p(t),v> . Therefore condition (1) of Corollary (2) i s s a t i s f i e d . Q.E.D. The following r e s u l t i s obtained by applying the "Hamiltonian-Jacobi" c r i t e r i o n (Theorem 5) to the generalized problem of Bolza (P c) corresponding to the optimal co n t r o l problem (C) . Suppose we are given an arc x . The following assumptions are made: (H^) there e x i s t s a p o s i t i v e number e such that f(x,u) 2 and g(x,u) are C on { (x,u) : x - x ( t ) < e, u e U} , 137 and the function Z° i s on {x: |x - x(b)| < e} . D e f i n i t i o n : Suppose that the Hamiltonian H i s C*~+ near an arc z = (x,p) . We say that the extended Jacobi condition i s s a t i s f i e d at z i f there e x i s t s a L i p s c h i t z matrix function Q(') on [a,b] such that, for a l l t i n [a,b] , Q(t) i s symmetric and s a t i s f i e s : n(t) - Q ( t ) T ( t ) + Q(t)B(t) + <5(t)Q(t) - a(t) > 0 for a l l t i n [a,b] , for a l l matrices a(t) 6(t) B(t) Y(t) e 3VH(z(t)), and f o r a l l n(t) e 3Q(t) The notations 3VH(z(t)) and 3Q(t) designate r e s p e c t i v e l y the generalized Jacobian of the L i p s c h i t z functions VH(') and Q(*) at z (t) and t . Theorem 8: Let (x,u) be an admissible p a i r f o r (C) . Assume that hypothesis (H^) holds f o r some p o s i t i v e e , and (1) U i s a nonempty convex compact polyhedron i n lR m ; (2) there e x i s t s an arc p on [a,b] s a t i s f y i n g f o r a l l t e [a,b] - P(t) = ( f x ( x ( t ) , u ( t ) ) ) T p ( t ) - g x ( x ( t ) ,u(t) ) , 138 with p(b) = - 1° (x (b)) (3) f o r a l l t e [a,b] and f o r a l l u e U such that u f u ( t ) , <p(t) ,f (x(t) ,u(t) )> - g( x ( t ) , u ( t ) ) > <p(t),f(x(t),u)> - g(x(t),u) ; fo r a l l t e [a,b] ; (5) the extended Jacobi condition i s s a t i s f i e d by a matrix function Q such that Then the p a i r (x,u) provides a strong l o c a l minimum for (C) . Remark: Condition (2) of Theorem (8) i s i n f a c t the combination of conditions (1) and (3) of the maximum p r i n c i p l e (see section 1.3), where we assume that problem (C) i s . normal (X = 1) . Furthermore, conditions (3) and (4) of Theorem (8) are strengthened forms of condition (2) of the maximum p r i n c i p l e . Remark: As we s h a l l soon see, conditions (1), (3), and (4) of Theorem (8) imply that the Hamiltonian defined by (6.4) i s . Hence, the use of the generalized Jacobian 3VH(*) i n the extended Jacobi condition i s j u s t i f i e d . (4) g (x(t),u(t)) - D ((f (x(t) , u ) ) T p ( t ) ) uu u u u=u(t) > 0 Q(b) < 1° (x(b)) 139 Proof of Theorem (8): Consider the generalized problem of Bolza corresponding to the optimal c o n t r o l problem (C): (P c) minimize J c ( x ) = £(x(a),x(b)) + a L ( x ( t ) , x ( t ) ) d t over a l l arcs x from [a,b] to 3Rn , where the functions £ and L are defined by (6.5) and (6.6) r e s p e c t i v e l y . Define J(x,p,q) = sup{<q,u> +<p,f(x,u)> - g(x,u): u e U} . Since U i s compact and f and g are continuous for x near x ', the function J(x,p,q) s a t i s f i e s the boundedness condition f o r x near x . Also, condition (3) of t h i s theorem and equation (6.6) imply L ( x ( t ) , x ( t ) ) = g ( x ( t ) , u ( t ) ) a.e. , and hence J c ( x ) i s f i n i t e . Thus, by applying Theorem (1) we conclude that L i s measurable and that (x,u) i s a strong l o c a l minimum for (C) whenever x provides a strong l o c a l minimum for (P^) . To prove the optimality of x f o r (P^) we w i l l apply Theorem (5) . F i r s t , we show that u i s continuous on [a,b] . Given t e [a,b] , l e t ke any sequence i n [a,b] tending to t . Since U i s compact, for a subsequence of {t }, (u(t.)} tends to a l i m i t u' i n U . For notational s i m p l i c i t y , we denote t h i s subsequence again by ^ j - ^ • s u f f i c e s to show that u = u(t) . 140 The functions x, p, f, and g are continuous. Moreover, condition (3) y i e l d s <p(t.),f(x(t.),u)>-g(x(t.),u) < <p(t J ,f (x(t J ,u(t _.))> g(x(t.) ,u(t.)) 3 3 f o r a l l u e U . Upon taking the l i m i t we get <p(t),f(x(t),u)> - g(x(t),u) < <p(t) ,f (x(t) ,u')> - g(x(t),u') f o r a l l u e U . By condition (3), <p(t),f(x(t),•)> - g(x(t),*) i s maximum on U only at u(t) . Thus, u' = u(t) and u i s continuous. 1 Since u i s continuous, equation (6.1) implies that x i s C and hence, by condition (2), p i s also C"*" . Thus, by condition (3) and equation (6.6) , L ( x ( t ) , x ( t ) ) = g( x ( t ) , u ( t ) ) f o r a l l t e [a,b] . We want to show that condition (a) of Theorem (5) holds for a l l t i n [a,b] . Let v s TRn and t e [a,b] . Define A(t,v) = {ueU: x(t) +v = f (x(t),u)} 141 If A(t,v) i s empty, equation (6.6) implies that L ( x ( t ) , x ( t ) +v) = + °°. On the other hand, i f A(t,v) i s not empty, then the infimum i n (6.6) i s attained at some u i n A(t,v) . Hence, using condition (3), and the a d m i s s i b i l i t y of (x,u) , i t follows that L(x(t) ,x(t) + v) - L(x(t) ,x(t)) = g(x(t) ,u(t)) - g ( x ( t ) , u ) > <p(t) ,f (x(t) ,u) - f (x(t) ,u(t))> = <p(t) ,v> . Thus condition (a) of Theorem (5) holds for a l l t i n [a,b] . For the r e s t of the proof we need to e s t a b l i s h the following r e s u l t . Given any p o s i t i v e number B , we define N^(z) = {z e TRn x TR n : | z - z ( t ) | < B for some t e [a,b]} . Lemma 6.1: Assume that the hypothesis (H^) and conditions (1), (3), and (4) of Theorem (8) hold. Then there e x i s t s a p o s i t i v e number B(B<e) such that f o r a l l (x,p) e N (z) the supremum i n (6.4) i s p attained at a unique point u(x,p) . Furthermore, the function u(*,*) i s L i p s c h i t z and s a t i s f i e s u ( x ( t ) , p ( t ) ) = u(t) . Proof: Hypothesis (H^) and condition (4) imply that there e x i s t s a p o s i t i v e number 6(6 <e) such that for a l l (x,p) e N„(z) and for o a l l u e {ueU: Iu - u ( t ) I < 6} we have: 142 (6.8) g (x,u) - D ({f (x,u)) Tp) > 0 . uu u u Since U i s compact and the functions f(x,*) and g(x,») are continuous, the supremum i n (6.4) i s always attained f o r any (x,p) i n N(£,«>) . However, we want to show that there e x i s t s a p o s i t i v e number y such that the supremum i n (6.4) i s attained at a unique value u(x,p) f o r a l l (x,p) i n N^(z) . Suppose that our claim i s f a l s e . Then, f o r any p o s i t i v e integer k we can f i n d elements fck' P k ' X k ' U k a n d U k S U C h t h a t t k e [a,b], ( x k'P k> e { ( x , p ) : jx - x ( t f c ) | <^ , |p - p ( t R ) | < ^  } , u° e U , u^ e U , u° ^ u^ , and JC TS. T\\ K. (6.9) <p k,f(x k,u)> - g(x k,u) < <p k,f(x k,u°)> - g(x k,u°) = <p k,f (x k,u k)>- g(x k,u k) fo r a l l u e U . The compactness of U and [a,b] implies t h a t there e x i s t s a subsequence of {(t ,u°,u^)} which converges to (t j U 0 , ^ ) e [a,b] xtJxU. For notational s i m p l i c i t y , we denote t h i s subsequence again by { (t ,u°,uhl . By the continuity of x and p , K K K {x, } • x ( t ) and {p, } > p (t ) . k o k o Upon taking the l i m i t i n (6.9) we get <p(t ) , f ( i ( t ),u)> - g ( x ( t Q ) , u ) < <p(t ),f(x(tQ),u°)> - g(x(t Q),u°) = <p(t ) , f ( x ( t ) , u 1 ) > - g ( x ( t Ju 1) o o o fo r a l l u e U . Using condition (3) we obtain u = u = u (t ) , o and hence {u°} y u ( t ) k o and {u,1} y u(t ) . k o Since u(*) i s continuous, we can f i n d a p o s i t i v e integer N such that f o r k > N we have | u ° - u ( t k ) | < 6 and where 6 i s the constant obtained e a r l i e r . Choose an integer N such that N > N and -7- < 6 . Thus, using ^ o o N o 144 (6.9), i t follows that the maximum of the function < P N ' f ( X N ' U ) > " g ( X N ' U ) o o o over the set {u e U: | u - u ( t N ) | < 6} i s attained at both points o u° and u"'' with u° 4 . We also have: N N N N o o o o o o o o ( x N ,P N ) e { ( x , p ) : | x - x ( t N )| < A- (<6), | p - P ( t N )| < i (<6)} o o o o o o However, r e l a t i o n (6.8) implies that the function given above i s s t r i c t l y concave i n u on {ueU: | u - u ( t N )| < 6} , and hence the maximum o cannot be attained at two d i f f e r e n t points. Thus, there e x i s t Y > 0 (-y < 8) and a unique function u(x,p) such that f o r (x,p) e N^(z) (6.10) H(x,p) = sup{<p,f(x,u)> - g(x,u): ueU> = <p,f(x,u(x,p))> - g(x,u(x,p)) Moreover, the uniqueness of the function u(x,p) s a t i s f y i n g (6.10) implies that u(',*) i s continuous. Also, by condition (3), u(x(t) ,p(t)) = u(t) . It remains to show that u(*, #) i s L i p s c h i t z . From (6.10) i t follows that f o r (x,p) e N (z) , Y (6.11) 0 e g u(x,u(x,p)) - <p,f u (x,u (x,p)) > + 3f I J(u(x,p)) , 145 where 3¥ (u ) i s the normal cone to U at u , that i s , U o o 3^ (11 ) = i {y e 3Rm ; <y ,u - u > < 0 f o r u e u} i f u e U o o i f u i U o Let X^(x,p) X m(x,p) be the eigenvalues of the matrix function g (x,u(x,p)) - D ((f (x,u(x,p))) p) , uu u u and define r 1 X(x,p) = min {X.(x,p)> . i = l 1 Using (6.8), we conclude that f o r a l l (x,p) e N (z) and f o r a l l Y h e 3Rn , X(x,p) > 0 and h. [g (x,u(x,p)) - D u( (f u(x,u(x,p)).) p)]h > X(x,p)|h Hypothesis (H^) implies that, f o r a l l i = 1,•••,m , X^(',*) i s continuous and so i s X(*,*) . Thus, there e x i s t positive'numbers u and g(g<6) such that f o r a l l (x,p) e N (z) and f o r a l l h e 3Rn 146 we have: (6.12) h. [g (x,u(x,p))-D ((f (x,u(x,p) )) Tp)]h >y|h| 2 . uu u u 1 1 We have that U i s a convex compact polyhedron, hypothesis (H^) i s s a t i s f i e d , and r e l a t i o n s (6.11) and (6.12) hold f o r a l l (x,p) i n N (z) . Then, Theorem (4.-2) and Coro l l a r y (4.3) of [18] imply the p f o l l o w i n g : There e x i s t s a p o s i t i v e number X such that f o r any y = (x ,p ) i n N„(z) we can f i n d a neighborhood N(y ) of y o o o B o o with (6.13) |u(z) -u(y ) I < xlz-y I for a l l z e N(y ) . 1 o 1 ' o • o This proves that u(«) i s uniformly p o i n t - L i p s c h i t z on N (z) . p However, we want to show that u(') i s L i p s c h i t z . In other words, we want (6.13) to hold f o r a l l p a i r s (v,w) i n N (z) . p * w — v Let v and w be vectors i n N. (z) , and l e t d = r . B |w - v I Define [v,w] = {z=v + t d : 0 < t < |v-w|} . Then, [v,w] c N (z) and hence, for each y e [v,w] there e x i s t s a P o neighborhood N ( y Q ) o f YQ such that f o r a l l z i n N ( y o ) r e l a t i o n (6.13) holds. Since [v,w] i s compact, we can f i n d vectors z = v, z,.•••.z , = w i n [v,w] and neighborhoods N(z ),«**,N(z ) o 1 r+1 o r+i 147 r+1 of z .•••,z , such that [v,w] c u N(z.) and o r+1 i = 0 i |u(y) - u(z J | < X |y - z^| f o r a l l y e N ( z J As elements of [v,w], the vectors z (i=0,•••,r+l) have the form z . = v + t. d , 1 x where 0 < t. < Iv - wI . x ' Rearrange z .•••,z , so that we have o r+1 0 = t < t, • • • < t . = v - w , o 1 r+1 1 1 and choose v ,v,,•••,v i n [v,w] such that o 1 r v. e N(z.) n N(z. ,) for a l l i = (),••••,r x x x+1 Thus, we obtain: I u (v) - u (w) < I u (v) - u (v ) + u (v ) - u (v. ) + 1 1 1 o ' 1 o 1 ' + |u(v ) - u(w) I r < X (I v - v l + |v - v . I +•••+ Iv -w|) i o 1 1 o 1 1 1 r 1 = X [ v - w | . .. Therefore, the function u(*) i s uniformly L i p s c h i t z on N (z) p Q.E.D. 148 Now, l e t us continue the proof of Theorem (8). F i r s t , we observe that the problems (C) and (P c) have the same Hamiltonian. By Lemma (6.1), there e x i s t a p o s i t i v e number g(B <e) and a unique L i p s c h i t z function u(x,p) on N (z) such that p H(x,p) = <p,f(x,u(x,p))> - g(x,u(x,p)) and u( x ( t ) , p ( t ) ) = u(t) . Hence, the Hamiltonian H(«,*) i s continuous and hypothesis (H^) holds. Moreover, using [3, Theorem (2.1)] and hypothesis (H^) i t follows that H(-,«) i s C 1 on N (z) with gradient p (6.14) VH(x,p) = (pf x(x,u(x,p)) - g x(x,u(x,p)),f(x,u(x,p))) . But the function u(»,') i s L i p s c h i t z on N (z) and hypothesis (H ) P 3 holds. Thus, (6.14) implies that H(«,') i s C 1 + on N (z) and P hypothesis (H^) i s s a t i s f i e d . The a d m i s s i b i l i t y of (x,u) , condition (2) and equation (6.14) imply that condition (b) of Theorem (5) holds. Also, condition (c) of Theorem (5) follows from condition (5) of t h i s theorem. Thus, the proof w i l l be completed when we prove that condition (d) of Theorem (5) holds. Since £°(«) i s C 2 near x(b) and p(b) = - £°(x(b)) , x then f o r x e {x: | x - x ( b ) | < e} w e have: 149 (6.15) £°(x) = £°(x(b)) - <p(b) ,x-x(b)> + i - <x - x(b) ,£° (x(b)) (x-x(b))> + X - x(b) \0 (x - x(b)) , 2 xx 1 where . . o (x - x(b)) l i m - = 0 x+x(b) | x - x ( b ) | n Let A = min {A.} , where A, , , , ,,A are the eigenvalues of the matrix . , l I n i = l £° x(x(b)) - Q(b) . By condition (5), t h i s matrix i s p o s i t i v e d e f i n i t e . Hence, A i s p o s i t i v e and f o r a l l d i n 3Rn <d, (£° x(x(b)) -Q(b))d> > A|d| 2 . Choose a p o s i t i v e number a such that f or a l l d s a t i s f y i n g I cl I < a we have : Thus, using (6.15), we obtain f or a l l such d £°(x(b)+d) - £°(x(b)) + <p(b),d> - . i - <d,Q(b)d> = j <d, (£° x(x(b)) -Q(b))d> + |d| 0(d) 150 . Therefore, condition (d) of Theorem (5) holds, where £ i s given by (6.5) . Q.E.D. Remark: For autonomous optimal c o n t r o l problems we have already shown that the Hamiltonian H , defined by (6.4), i s C^"+ . However, f o r a large c l a s s of optimal c o n t r o l problems the Hamiltonian H i s not 2 nece s s a r i l y C (see the example given below). This f a c t demonstrates the u t i l i t y of considering the Hamiltonian to be which i s l e s s 2 r e s t r i c t i v e than C Remark: Consider the optimal c o n t r o l problem where f and g depend e x p l i c i t l y on t . If we assume that the c o n t r o l function u(t) i s i n the i n t e r i o r of the c o n t r o l set U , then the Hamiltonian H(t,z) 2 i s C i n z . For t h i s s p e c i a l case, a s u f f i c i e n c y theorem s i m i l a r to Theorem (8) can be stated, where the assumption that U i s a convex polyhedron (which was needed to prove that H i s C^"+) can be omitted. I f , i n addition, we assume that the function Q of condition (8) i s C^ " , Lemma (5.1) implies that the i n e q u a l i t i e s i n condition (5) can be replaced by e q u a l i t i e s . This s p e c i a l case has been considered by Mayne [15, Theorem (3.2)]. Remark: For the case of optimal c o n t r o l problems where both boundary values are f i x e d (x(a)=A, x(b) = B) , Theorem (8) remains v a l i d , where the boundary conditions p(b) = - £°(x(b)) and Q(b) < 1° (x(b)) X XX 151 are removed. Example: Consider the following optimal c o n t r o l problem where both boundary values are f i x e d : 1 (C) minimize 3 (2uJ(t) + 2 u 2(t) - dt 0 subject to x(t) = x(t) (u (t) +u (t)) x(0) = x ( l ) = 1 and ( u x ( t ) , u 2 ( t ) ) e U = [-1,0] x [-1,0] for a l l t e [0,1] For t h i s problem we have: g(x,u) = 2 U ; L + 2u 2 - — f(x,u) = x(u + u 2) Consider x(t) = 1 and u(t) = 0 f o r a l l t e [0,1]. We intend to apply Theorem (8) to prove that (x,u) = (1,0) i s a strong l o c a l minimum f o r (C). C l e a r l y (x,u) = (1,0) i s an admissible p a i r for (C) and 2 the functions g and f are C . Also, the set U = [-1,0] x [-1,0] 152 i s a convex compact polyhedron. Furthermore, Define f (x,u) = u. + u. , f (x(t),u(t)) = 0 , x 1 2 x x ~ 1 g (x,u) = - — , and g (x(t),u(x)) = - — . x 8 x 8 Then, p(t) = ^ — - for a l l t e [0,1] - p(t) = - = f x ( x ( t ) ,u(t))p(t) - g x ( x ( t ) ,u(t)) Hence condition (2) of Theorem (8) holds. Moreover, g (x(t),u(t)) - D (f (x(t) ,u)p(t)) -uu u u 'u=u(t) 4 0 0 4 for a l l t e [0,1] , which implies that condition (4) of Theorem (8) i s s a t i s f i e d . The Hamiltonian corresponding to t h i s problem i s : 2 x~ H(x,p) = sup{px(u +u 2) - 2u 1 - 2u 2 + ^ : ( u l f u 2 ) e U> . 2 2 x Since the function px(u^ i u ^ ) - 2u^ - 2u 2 + — i s s t r i c t l y concave i n u , the supremum above i s attained at a unique value u(x,p) 153 which can be computed by using the method of Lagrange m u l t i p l i e r s . Thus, (6.16) u(x,p) = { px px • 4 ' 4 (0,0) (-1,-1) i f px e [-4,0] i f px > 0 i f px < - 4 , and (6.17) H(x,p) = i 2 2 px_ x 24 24 - 2px - 4 + x 24 i f px e [-4,0] i f px > 0 i f px < -4 . Since x(t) = l , p(t) 1 - t , then p ( t ) x ( t ) > 0 f o r a l l t in [0,1] and hence, from (6.16) i t follows that the maximum of p(t) f ( x ( t ) , u ) - g(x(t),u) i s attained only at u(x(t) ,p(t)) = (0,0) = u(t) . Thus, condition (3) of Theorem (8) holds. Equation (6.17) implies that the Hamiltonian H(x,p) i s continuously d i f f e r e n t i a b l e with gradient VH given by 2 2 2 3 X X P X . '~2~ + ~Q~ i ^Y~) i f px e [-4,0] 2 (6.18) VH(x,p) = i ( ^ ,0) i f px > 0 ("2p +-Q- , -2x) i f px < -4 . 154 Hence, for x(t) = 1 and f o r p(t) = 1 0 t we have: H (x(t),p(t)) = ±-xx 4 for a l l t e [0,1] , H p x ( x ( t ) ,p(t)) = H (x(t) ,p(t)) = 0 for a l l t e [0,1], and H (x(t),p(t)) = \ does not e x i s t i f t f 1 i f t = 1 Therefore, the Hamiltonian H(x,p) i s not near z = (x,p) . However, since conditions (1), (3), and (4) of Theorem (8) are s a t i s f i e d , Lemma (6.1) implies that the function u(x,p) i s L i p s c h i t z near (x,p) and so i s the gradient VH(•,••) . I t remains to show that the extended Jacobi condition holds. By [8, Section 15], the following generalized Jacobian i n c l u s i o n holds : 3VH(x,p) c A(x,p) = 3 H (x,p) 3 H (x,p) x x p x 3 H (x,p) 3 H (x,p) x P P P Thus, to check the extended Jacobi condition for the elements of 3VH(x(t),p(t)) i t s u f f i c e s to check t h i s condition f o r the elements of the l a r g e r set A(x(t),p(t)) . 155 Equation (6.18) implies that 3 H ( x ( l ) , p ( D ) = [0, p p 2 and hence, A(x(t),p(t)) = 1 4 3 H (x(t),p(t)) p p where 3 H (x(t) ,p(t)) = P P [0, \\ i f t ^ 1 i f t = 1 Consider Q (t) = t on [0,1] , then Q (*) i s c , a n d f o r a l l t e [0,1] and for a l l a 6 3 Y A(x(t),p(t)) we have: Q (t) - YQ 2(t) + Q (t)6 + 6Q \t) - a = f - Y t 2 , o o o o 4 where y e [0,-j] However, 156 J - Y t 2 > ^  (>0) for a l l t e [0,1] . Thus, condition (5) of Theorem (8) holds. Using Theorem (8) we conclude that (x,u) i s a strong l o c a l minimum f o r (C). Remark: In the example given above the co n t r o l function u(t) i s i d e n t i c a l l y zero on [0,1]. Hence, u(t) i s on the boundary of the co n t r o l set U = [-1,0] x [-1,0] . Therefore, the s u f f i c i e n c y theorem given i n [15] cannot be employed f o r our example. Moreover, the Hamiltonian H(x,p) given by (6.17) i s not concave-convex, but rather j o i n t l y convex. Thus, the convexity approach (Chapter 2) also cannot be used. 157 Bibliography [1] G.A. B l i s s , Lectures on the Calculus of Va r i a t i o n s , University of Chicago Press, Chicago (1946). [2] V.G. B o l t y a n s k i i , S u f f i c i e n t conditions f or optimality and the j u s t i f i c a t i o n of the dynamic programming method, SIAM J . Control, (1966), pp. 326-361. [3] F.H. Clarke, Generalized gradients and a p p l i c a t i o n s , Trans. Amer. Math. S o c , 205(1975), pp. 247-262. [4] , Admissible r e l a x a t i o n i n v a r i a t i o n a l and c o n t r o l problems, J . Math. Anal. Appl., 51(1975), pp. 557-576. [5] , Mimeo, Univ e r s i t y of B r i t i s h Columbia, Vancouver, B.C., 1975. [6] , Necessary conditions f or a general c o n t r o l problem, i n Proceedings of the Symposium on the Calculus of Variations and Optimal Control, D.L. Russell, Editor (Mathematics Research Center, University of Wisconsin-Madison), Academic Press, N.Y. (1976). [7] , The general problem of Bolza, SIAM J. Control Optimization, 14(1976), pp. 682-699. [8] , Generalized grandients of L i p s c h i t z f u n c t i o n a l s , Tech. Report #1687(1976), Mathematics Research Center, Madison; Advances i n Mathematics, 40(1981), pp. 52-67. [9] , Extremal arcs and extended Hamiltonian systems, Trans. Amer. Math. S o c , 231(1977), pp. 349-367. [10] W.H. Fleming and R.W. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, N.Y. (1975). [11] I.M. Gelfand and S.V. Fomin, Calculus of Variations (translated by R. Silverman), P r e n t i c e - H a l l , Englewood C l i f f , N.J. (1963) . [12] P. Hartman, Ordinary D i f f e r e n t i a l Equations, John Wiley and Sons, New York, N.Y. (1964). [13] M.R. Hestenes, Calculus of Variations and Optimal Control Theory, Wiley (1966). [14] V.F. Krotov, Methods f o r s o l v i n g " v a r i a t i o n a l problems on the basis of s u f f i c i e n t conditions f or an absolute minimum, I, I I , I I I , Automat. Remote Control, (1962) pp. 1473-1484; (1963) pp. 539-553, (1964) pp. 924-933. 158 [15] D.Q. Mayne, S u f f i c i e n t conditions f or a co n t r o l to be a strong minimum, J . Opt. Theory Appl., 21 (1977) , pp. 339-351. [16] D.C. O f f i n , A Hamilton-Jacobi approach to the d i f f e r e n t i a l i n c l u s i o n problem, Thesis, University of B r i t i s h Columbia, Vancouver, B.C. (1979). [17] L.S. Pontryagin, V.G. B o l t y a n s k i i , R.V. Gamkrelidze, and E.F. Mishchenko, The Mathematical Theory of Optimal Processes, John Wiley and Sons, N.Y. (1965). [18] S.M. Robinson, Generalized equations and t h e i r solutions, Part I I : Applications to nonlinear programming, Preprint, University of Wisconsin-Madison, Madison, Wisconsin (1980) . [19] R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, N.J. (1970). [20] , Conjugate convex functions i n optimal c o n t r o l and the cal c u l u s of v a r i a t i o n s , J . Math. Anal. Appl., 32 (1970) , pp. 174-222. [21] , Generalized Hamiltonian equations for convex problems of Lagrange, P a c i f i c J . Math., 33(1970), pp. 411-427. [22] , Optimal arcs and the minimum value function i n problems of Lagrange, Trans. Amer. Math. S o c , 180(1973), pp. 53-83. [23] , Existence theorems for general c o n t r o l problems of Bolza and Lagrange, Advances i n Mathematics, 15(1975), pp. 312-333. [24] A. Seierstad and K. Sydsaeter, S u f f i c i e n t conditions i n optimal c o n t r o l theory, Inter. Econo. Rev. J., 18(1977), pp. 367-391. [25] R.B. Vinter and R.M. Lewis, The equivalence of strong and weak formulations f or c e r t a i n problems i n optimal c o n t r o l , SIAM J. Control Optimization, 16(1978), pp. 546-570. [26] , A necessary and s u f f i c i e n t condition of optimality of dynamic programming type, making no a p r i o r i assumptions on the c o n t r o l s , SIAM J . Control Optimization, 16(1978), pp. 571-583. [27] , A v e r i f i c a t i o n theorem which provides a necessary and s u f f i c i e n t condition for optimality, IEEE Trans, on Automatic Control, V o l . AC-25, No. 1 (1980). [28] L.C. Young, Lectures on the Calculus of Variations and Optimal Control Theory, Saunders, Philadelphia, PA (1969). 159 [29] V. Zeidan, S u f f i c i e n t conditions f or the generalized problem of Bolza, Trans. Amer. Math. Soc. (to appear). 

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