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A generalization of the Hamilton-Jacobi equation Havelock, David Ian 1977

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A GENERALIZATION OF THE HAMILTON - JACOBI EQUATION B.Sc, Carleton Un i v e r s i t y , Ottawa, 1974 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF THE FACULTY OF GRADUATE STUDIES i n the Department of MATHEMATICS We accept t h i s thesis as conforming to the THE UNIVERSITY OF BRITISH COLUMBIA September, 1977 Cc) David Ian Havelook, 1977 by David Ian Havelock MASTER OF SCIENCE In required standard In present ing th is thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for s c h o l a r l y purposes may be granted by the Head of my Department or by h is representa t ives . It is understood that copying or p u b l i c a t i o n of th is thes is for f inanc ia l gain sha l l not be allowed without my wr i t ten permission. Department of The Un ivers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 i i ABSTRACT Aft e r a b r i e f review df the relevant c l a s s i c a l theory and a presentation of the concept of generalized gradients, i t i s demonstrated that, i n analogy with the c l a s s i c a l case, a l o c a l l y l i p s c h i t z value function s a t i s f i e s a generalized version of the Hamilton-Jacobi equation. A s u f f i c i e n c y condition f o r optimality i s developed and some examples i l l u s t r a t i n g various aspects of the generalized theory are presented. i i i ACKNOWLEDGEMENTS I wish to acknowledge the patient and helpful assistance of my thesis supervisor, Frank Clarke, during the development of this thesis. I would also l ike to thank my Aunt Pauline for her unlimited hospital i ty and encouragement during my stay in Vancouver. David Havelock Ottawa September 29, 1977. iv TABLE OF CONTENTS Introduction 1 Chapter I - Preliminaries 6 - Equivalent Problems 8 - The Classical Hamilton -Jacobi Equation 9 - The Non-classical situation 12 - Lemma I Conditions For a Lipschitz Value Function 15 Chapter II - Generalized Gradients 20 Chapter III - The Generalized Hamilton - Jacobi Equation 26 - Theorem I Necessary Conditions. 28 - Lemma II Lower Bounds On The Value Function 36 - Lemma III Sufficient Conditions For The Value Function 38 - Theorem II Sufficiency Conditions For Optimality 40 Chapter IV - Examples 43 - Example I 43 - Example II 46 - Example III 47 - Example IV 53 Bibliography 55 1 INTRODUCTION The basic problem in the Calculus of Variations is that of finding a piecewise smooth curve y(x) which minimizes the definite integral j F(x,y(x),y'(x))dx and joins two fixed points (a ,b ) and (a,b). The integrand F is classically at least once continuously differentiable. The set of curves over which the minimum is sought is called the Set of Admissible Curves. Caratheodory ['2;P205,VOL III . took the approach of considering problems which were equivalent, in some sense, to a nice type of problem. A 'Nice' problem was said to be one for which the integrand F satisfies MIN F*x,y,q) = 0 q for a l l x,y . A problem with an integrand F i s said to be Equivalent to a 'nice' problem with an integrand F i f there exists a smooth function R(x,y) with F*(x,y,q) = F(x,y,q) - R'(x,y;l,q) where R'(x,y;l,q) is the directional derivative of R in the A direction (l,q). The definite integrals of F and F along any 2 admissible curve w i l l d i f f e r by the value R(a,b) - R(a ,b ) and o o hence we are assured that a curve w i l l solve the nice problem ( i e ; w i l l be optimal) i f and only i f i t solves the equivalent one. I t i s found that a problem i s equivalent to a 'nice' problem exactly where there e x i s t s a smooth s o l u t i o n to a c e r t a i n p a r t i a l d i f f e r e n t i a l equation c a l l e d the Hamilton-Jacobi Equation (H.-J.Eq.) H(x,y,R 2(x,y)) + R^x.y) = 0 where and a r e r e s p e c t i v e l y the f i r s t p a r t i a l d e r i v a t i v e s of R i n the f i r s t and second v a r i a b l e s . The Hamiltonian Function H(a,b,p) i s defined as H(a,b,p) = P ^ - F(a,b,qp) where q^ and p are r e l a t e d i m p l i c i t l y by the r e l a t i o n —F(a,b,q) | = p . 9q q=qp. ... The Value Function, or Hamilton's C h a r a c t e r i s t i c Function S i s given as a function of the end point (a,b): •a S(a,b) = MIN | F(x,y(x),y'(x))dx a with the minimum being over admissible curves from (a ,b ) to ° o o (a,b). Where the value f u n c t i o n i s defined and smooth, i t s a t i s f i e s the Hamilton - Jacobi Equation. This i n d i c a t e s that 3 many problems are equivalent to 'n ice ' problems. There is a greater variety of necessary conditions for optimality than of sufficiency conditions, but for 'nice ' problems we have a part icular ly simple sufficiency condition at hand: " i f F(x,y(x) ,y'(x)) = 0 almost everywhere along the curve y, then y i s optimal. " For problems equivalent to a 'n ice ' problem, the corresponding condition would require that F(x,y(x) ,y ' (x)) - R ' ( x , y ( x ) ; l , y ' ( x ) ) = 0' almost everywhere-along with R being some smooth solution to the H.-J .Eq. If the value function S is smooth along the optimal curve y then i t turns out that y necessarily sat is f ies the above relat ion with R = S. A major d i f f i cu l ty in applying this theory is the requirement of d i f f e rent i ab i l i ty . In many interesting cases there i s no guarantee that the value function w i l l be smooth (and i t often is not). The c las s ica l soap bubble problem, discussed in example III of chapter IV, i s such a case in which the value function actually f a i l s to be smooth. By employing the concept of Generalized  Gradients as defined for l oca l ly l ipschi tz functions, the"theory may be studied using functions which f a i l to be differentiable. Although the generalized gradient has been defined for a larger class of functions (see [3]) we w i l l only consider i t for loca l ly l ip schi tz functions here. For a loca l ly l ip sch i tz function f the generalized gradient at a point x, denoted 3f(x), i s a compact, convex, non-empty set. 4 If f i s a convex function df coincides with the s u b d i f f e r e n t i a l of f. Accompanying the generalized gradient i s the generalized d i r e c t i o n a l d e r i v a t i v e , denoted f°(x,y;a,b). L i k e the standard d i r e c t i o n a l d e r i v a t i v e f'(x,y;a,b), the generalized d i r e c t i o n a l d e r i v a t i v e i s a s i n g l e valued mapping. C l a s s i c a l theory presumes the existence of several minima and maxima which our generalized theory replaces with infima and suprema. We use a Generalized Hamiltonian H(x,y,p) = SUP ( pq - F(x,y,q) ) q to accomodate a greater v a r i e t y of integrands. This d e f i n i t i o n of H, u n l i k e that of the c l a s s i c a l Hamiltonian function, i s not predicated on the existence of the Legendre transform. I t i s found that 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 R to a Generalized Hamilton - Jacobi Equation (G.H.-J.Eq.) MAX ( H(x,y,v) + u ) = 0 (u,v)e8R(x,y) (where the 'MAX' e x i s t s automatically) establishes an equivalence with a generalized type of 'nice' problem: those s a t i s f y i n g INF F(x,y,q) = 0 . If the value function, as defined i n c l a s s i c a l theory, i s l o c a l l y l i p s c h i t z , i t i s found to s a t i s f y the G.H.-J.Eq. For the generalized theory, the value function i s defined for a l l (a,b) as J a S(a,b) = INF | F(x,y(x),y'(x))dx a 5 If the infimum of the value function i s not attained (no optimal curve e x i s t s ) then the value function may f a i l to s a t i s f y the G.H.-J.Eq., but w i l l s a t i s f y the following Generalized Hamilton-Jacobi Inequality  (G.H.-J.Ineq.) : H(a,b,v) + u ^ 0 fo r a l l (u,v) i n 8 S(a,b). A s u f f i c i e n t c o n d i t i o n f o r op t i m a l i t y i s provided by the solutions of the G.H.-J.Ineq. as follows: I f R i s a s o l u t i o n to the G.H.-J.Ineq. on a region and i s a curve l y i n g i n t h i s region and s a t i s f y i n g F(x,y o(x),y^(x)) - |^R(x,y o(x)) = 0 almost everywhere, then y^ i s an optimal curve (provides the infimum i n the value function) . 6 CHAPTER I Pr e l i m i n a r i e s By a curve we w i l l mean a l i p s c h i t z function mapping an i n t e r v a l [a ,a] of the r e a l l i n e 8. in t o the n-dimensional r e a l L o J space Rn. We f i x a point (a ,b ) i n 1 X 1 and henceforth consider o o only curves y which s a t i s f y y(a ) = b . Given a set U i n l n ^  we o o say a curve y l i e s i n U almost everywhere i f (x,y(x)) i s i n U for almost a l l x i n Ta , a l . o J Let F : R X _.n X fcn -> R. be a continuous functi o n , l e t U be n+1 n+1 a set i n 1 , and l e t (a,b) be a point i n K. D e f i n i t i o n The following problem i s r e f e r r e d to as a basic problem: f i n d INF j F(x,y(x),y'(x))dx yeA(a,b ) J a Q where A(a,b) := {curves y l y i n g i n U almost everywhere with y(a) = b}. I t i s i m p l i c i t that y(a )•= b Q and that y i s l i p s c h i t z (hence y' e x i s t s almost everywhere on [a ,al) . Notice that i t i s assumed that a > a . o o For b r e v i t y , where no confusion r e s u l t s , F(x,y(x),y'(x)) w i l l be written as F(x,y,y'). The basic problem i s characterized by three things: the function F; the set U; and the point (a,b). The set U w i l l be r e f e r r e d to as the domain, the point (a,b) as the terminal point, and the function F as simply the integrand. The set A(a,b) w i l l be r e f e r r e d to as the set of admissible curves to (a,b). Notice that the point (a,b) must l i e i n the closure of the domain U, U. 7 For s i m p l i c i t y of presentation we w i l l henceforth consider only one-dimensional problems, that i s ; R n = Ht"*", wherein curves w i l l map BL into R. Consider the family of basic problems determined by a f i x e d integrand F, a f i x e d domain U, and a set of terminal points ft. We 2 w i l l require that as a subset of R, ft be an open set and we w i l l r e f e r to ft as the set of termination for the family of problems. D e f i n i t i o n For the family of problems described above, we define the.. value function S on ft as follows: f o r (a,b) i n ft S(a,b) = INF f F(x,y,y')dx yeA(a,b) J a If the infimum i s a t t a i n e d by some y i n A(a,b), we say that y i s an optimal curve to (a,b) or y i s optimal i n A(a,b). Furthermore, i f y i s an admissible curve to (a,b) s a t i s f y i n g a F(x,y,y')dx <_ S(a,b) + 6, (for 6>0) a o then we say that y i s a 6-near optimal curve to (a,b). A sequence r l ° ° of curves {y 1 _, with each y a 6 -near optimal curve to (a,b) with n n-1 n n 6 n 0 as n ->• 0 0 w i l l be c a l l e d a minimizing sequence. Notice that as long as S(a,b) i s f i n i t e , there must be a minimizing sequence i n A(a,b) Equivalent problems * Two integrands F and F are s a i d to be equivalent on a set 2 ( U say ) i n 1 i f there e x i s t s a f u n c t i o n R which i s continuously d i f f e r e n t i a b l e on U and s a t i s f i e s VR(a,b) • (l,q) = F(a,b,q) - F*(a,b,q) for a l l (a,b) i n U, and a l l q i n R. The symbol V represents the usual vector gradient while the symbol • represents the usual inner 2 (scalar) product on R . Consider the two basic problems INF f F(x,y,y')dx yeA ( a , b ) J a and INF j F'(x,y,y')dx yeA(a,b)J a o * " with F and F equivalent on the domain U. Since a * j F(x,y,y»)dx - j a * | F (x,y,y')dx a o VR(x,y(x)) • (l, y ' ( x ) ) d x a = R(a,b) - R(a ,b ) , o o a curve y i n A(a,b) w i l l be optimal or 6-near optimal for one problem i f and only i f i t i s optimal or 6-near optimal r e s p e c t i v e l y f o r the ; other. Accordingly, the two problems are s a i d to be equivalent problems. 9 A basic problem or i t s integrand F i s called nice ( on the domain U ) i f • MIN F(x,y,q) = 0 qe l for a l l (x,y) in U. Notice that when a basic problem i s nice, i t is suff icient (although not necessary) for y to be optimal in A(a,b) to have F (x ,y ,y ' ) = 0 for almost a l l x in [a^.a]. The concepts of equivalent problems and nice problems were introduced by Caratheodory ( [2 ;§227 vol II] ). In their c la s s ica l setting i t is assumed that both the integrand F and the function R are smooth. In chapter III we w i l l alter the definitions s l i gh t ly , freeing us from these smoothness assumptions. The Class ica l Hamilton-Jacobi Equation Let F(x,y,q) be a twice continuously differentiable integrand a 2 with Y~2 F(x>y>9) — 0 f ° r a H 9 i n The mapping from {(x,y)} X R into {(x,y)} X 1 given by is one to one (although the range may be a proper subset of {(x,y)} X I ) The c la s s i ca l Hamiltonian function H i s defined as H(x,y,p) = p • qQ - F(x,y,qQ) where q i s determined impl i c i t ly by the relationship a_ 3'q F ( x , y , q ) L = q=qo P 10 The transformation df the system (x,y,q,F) into the system (x,y,p,H) is known as a Legendre Transform [6,§7.1], Notice that H(x,y,p) may not be defined for a l l p in IL. Assume that H is defined at the point (x,y,p) and consider the following function in q: fCq) = pq - F (x ,y ,q ) . Since this function is concave and the f i r s t derivative vanishes at q o g ( p ——F(x,y,q ) = 0 ), we must have a maximum occuring at q . dq o o Consequently we see that where H(x,y,p) i s defined H(x,y,p) = MAX (pq - F(x,y,q) ) . qel Furthermore, notice that the Hamiltonian w i l l be defined exactly where the maximum in the above expression exists . Suppose that R establishes an equivalence between F and a nice integrand F on U, then R*(x,y,q) = F(x,y,q) - VR(x,y)- (l ,q) for (x,y) in U and q in IL. It follows that MAX (-F*(x,y,q) ) q = MAX ( | | + | | -q " F(x,y,q) ) 0, and so R sat is f ies the part ia l d i f ferent ia l equation 11 H( x , y , f^ R(x,y) ) + f^ R(x,y) = 0 on U. This equation i s c a l l e d the Hamilton-Jacobi Equation (H.-J.Eq.). Notice that conversely, i f R s a t i s f i e s the H.-J.Eq. on U then i t provides an equivalence between F and the n i c e integrand F*(x,y,q) = F(x,y,q) - VR(x,y)-. (l,q) on U. Caratheodory's lemma [7,theorem 5.1] asse r t s that i f the infimum of the value function i s attained at every point of an open set V and i f i t i s continuously d i f f e r e n t i a b l e on V, then the value function establishes an equivalence between F and a nice integrand on V. Consequently, the value function w i l l s a t i s f y the H.-J.Eq. on the open set V. 12 The Non-Classical Situation For integrands which may not satisfy the class ical assumption of being twice continuously di f ferentiable , we define the more versat i le generalized Hamiltonian H(x,y,p) = SUP (pq - F(x,y,q) ) . qeR As a supremum of affine functions in p the generalized Hamiltonian w i l l be convex in p, and may assume values of plus i n f i n i t y . Unlike the c las s ica l Hamiltonian i t may happen that the supremum is not attained by any q in R, or that i f the supremum is attained i t i s attained for more than one q. Whenever the assumptions that F i s twice continuously differentiable and s t r i c t l y convex in q are v a l i d , the c la s s ica l Hamiltonian, where i t is defined, w i l l equal the generalized Hamiltonian. Without confusion, then "Hamiltonian" w i l l henceforth refer to the generalized Hamiltonian. Consider the H.- J .Eq . in which the generalized Hamiltonian i s employed. If R is a continuously differentiable solution to this equation, then setting F (x,y,q) = F(x,y,q) - VR(x,y) • ( l ,q) gives INF F*(x,y,q) q = -SUP [ VR(x,y)- ( l ,q) - F(x,y,q) ] q = - ~ R ( x , y ) - H( x , y , -^RCx.y) ) = 0 . 13 Unlike the s i t u a t i o n i n the c l a s s i c a l case, we are not assured here that the infimum i s a c t u a l l y a t t a i n e d . In order to extend the c l a s s i c a l theory then, i t seems n a t u r a l to modify the d e f i n i t i o n of n i c e problems or integrands as follows: DEFINITION A basic problem or i t s integrand F i s s a i d to be n i c e on U if_ INF F(x,y,q) = 0 q for a l l (x,y) jLn U. The c l a s s i c a l theory demands not only that the integrand be twice d i f f e r e n t i a b l e , but also that ' ( i ) the value function be continuously d i f f e r e n t i a b l e and ( i i ) optimal curves e x i s t ( i e : the infimum def i n i n g the value function i s always attained ) (see [7,§3] ). These two c o n s t r a i n t s on the value f u n c t i o n are d i s t i n c t from smoothness and convexity i n q imposed on the integrand. As we see i n example I I I chapter IV, the assumptions on the integrand are s a t i s f i e d , however those on the value function are not. If the integrand need not be convex i t i s very easy to construct simple examples i n which optimal curves do not e x i s t . Consider f o r example, F(x,y,q) = (1 + q 2 ) " 1 . 14 U s i n g "sawtooth shaped" c u r v e s we can g e t j F(x,y,y')dx J a o as c l o s e to z e r o as we w i s h , y e t s i n c e F i s s t r i c t l y p o s i t i v e , no c u r v e w i l l p r o v i d e t h e infimum v a l u e o f z e r o . I f the i n t e g r a n d need n o t be smooth, i t i s easy t o c o n s t r u c t examples i n which the v a l u e f u n c t i o n i s n o t smooth. The s i m p l e i n t e g r a n d F ( x , y , q ) = |q| f o r example, y i e l d s t h e non-smooth v a l u e f u n c t i o n S (x,y) = |yj . S t u d i e s have been made of i n t e g r a n d s which may f a i l t o be smooth but r emain convex, o r l i p s c h i t z (see [5] and the r e f e r e n c e s p r o v i d e d t h e r e ) . A l t h o u g h the v a l u e f u n c t i o n f r e q u e n t l y f a i l s t o be smooth, the a u t h o r knows of no c a s e i n which the v a l u e f u n c t i o n i s f i n i t e b u t n o t l o c a l l y l i p s c h i t z as l o n g as the i n t e g r a n d remains c o n t i n u o u s . In f a c t , i f F i s l i p s c h i t z and i t happens t h a t we can f i n d a r e a l K such t h a t f o r a l l (a,b) i n a neighbourhood 17 , t h e o p t i m a l c u r v e y , t o (a,b) ab s a t i s f i e s |y'| <K, t h e n t h e v a l u e f u n c t i o n S w i l l be a s s u r e d of b e i n g l i p s c h i t z i n the n e i g h b o u r h o o d . The bound K can be g u a r a n t e e d by growth c o n d i t i o n s on the i n t e g r a n d . 15 LEMMA I CONDITIONS FOR A LIPSCHITZ VALUE FUNCTION Let Q, be a set of termination of the form n = { ( x , y ) e x > e > 0, and | a p - x| + |b - y < M < 0 0 } for fixed e,M i n R, and l e t the domain U be R . The value function S(a,b) = INF I F ( x , y , y ' ) d x yeA(a,b) / a J o w i l l be l i p s c h i t z on Q i f there e x i s t p o s i t i v e constants and 3 3 such that for a l l (x,y,q) e R and (a,8,Y) e R , F s a t i s f i e s ( i ) F(x,y,q) _> 1 and ( i i ) F(x+cx,y+g,q+Y) 1 F(x,y,q) • exp ! I (a ,e,Y)| I (k2+|I(x,y,q)|j) Condition ( i ) requires only that F be bounded below, since the addition of a constant to the integrand merely adds a l i n e a r term to the value function. Condition ( i i ) i s a growth condition which, for d i f f e r e n t i a b l e functions, assumes the form k. f ' ( x ) < f ( x ) -x +k. Functions of the form f(x) = jxj for a >1 s a t i s f y t h i s condition 1 but functions with cusps pointing downward, such as f(x) = |x| 2 do not. 16 Proof of Lemma 1 We begin by showing that S i s bounded on ft. Assume, without loss of generality, that (a o > b Q ) = (0,0). For (a,b) in ft we-consider the straight l ine y e A(a,b) given by y(x) = bx/a. S(a,b) <_ f F(x,bx/a,b/a)dx J 0 <j F(0,0,0)- ex P [ kx| | (x,bx/a,b/a)|| /k2 ]dx < J F(0,0,0)- exp[ ^1 | (M,M,M/e)| |/k2 ]dx < M- F ( 0 , 0 , 0 ) « exp[ kJ | (M,M,M/e)|| /k2 ] < °° • Thus S has an upper bound and a lower bound as desired (the lower bound being zero, since F is pos i t ive) . To show that S i s l ip sch i tz on ft, i t is sufficient to show that i t is l ipschi tz separately in the variables x and y. Let y^ and y^ be 6-near optimal curves to (a^,b^) e ft and (a^ b^) e ft, respectively. Define the curve z e A(a^,b^) by z(x) = Y 2(x) + (b 1 ~b 2 )x/a 1 . Now, 17 S(a 1 ,b 1 ) < J 1 F ( x , z ( x ) , z , ( x ) ) d x k.| | ( 0 , ^ X ) b i - b 2 , .1 ai &i J%(x,y 2 ,y 2 j 1 F ( x , y 2 , y 2 ) -) • exp exp ) • exp I I (x , y 2 , y ' 2 ) l| -Hk2 b1 w ^ - ^ i r * ) ! a i dx dx dx . From elementary calculus, i f x i s bounded then 3K e & such that e l x l £ 1 + [Kx|. Using this fact we see that S (a 1 ,b 1 ) < J 1 F ( x , y 2 , y 2 ) d x . [ 1 + 'Ki b ^ l ] < [ 6 + S (a l 5 b 2 ) ] • [ 1 + K | b 1 - b 2 | ] Rearranging and le t t ing 6 0, S (a l 5 b ) - S(a 1 ,b 2 ) < K- |b1-l>21 where K can be chosen independently of a^, b^, b^-Interchanging b^ and b 2 in the above argument we get S (a l S b 2 ) - S(a 1 ,b 1 ) < K- |b 1 ~b 2 | , and so S is l ip sch i tz in the second variable. To see that S i s l ip sch i tz in the f i r s t variable, we redefine the curves y^ and y 2 to be 6-near optimal curves to (a^,b^) E ft and 18 (a ^ j b ^ ) e ft, r e s p e c t i v e l y . L e t u(x) = a^x/a^ and d e f i n e z e ACa^b^) as z ( x ) = y^CuCx)). We have, S ( a 1 , b 1 ) < j 1 F ( x , z , z ' ) d x 2 J 0 / a 2 ' y 2^u- ) ' a 2 y 2 ^ u ^ a i ^ d u * exp k I I ( - a-1^2 ( ^ 2 U , o , ± i l i 2 y ' )|| u , 0a 2 a j du k 9 + | | ( u , y 2 , y 2 ) | | a l f a ? 2 J 0 F ( u , y 2 ' y 2 ) " e x p f k l ' ( I a l " a 2 l / a2 + I a i - a 2 f / a l >3 du. As before, the exponential term can be bounded by K.|a^-a,J and we get S ( a 1 , b 1 ) < J 1 F ( x , y 2 , y 2 ) d x . [ 1 + K ^ - a^J ] (a.K + 1) [ 6 + S ( a 2 , b 1 ) ] • [ 1 + — ± • |a r a 2 l i Rearranging and l e t t i n g 6 -> OJ S ( a 1 , b 1 ) - Sia^b^ < K-w i t h K independent of a^, a 2 > and b^. As b e f o r e , by interc h a n g i n g a^and a 2 we o b t a i n S ( a 2 , b 1 ) - S ( a 1 , b 1 ) < K- la^-a^ , 19 and so S i s also l i p s c h i t z i n the f i r s t v a r i a b l e . This completes the proof. 20 Chapter I I GENERALIZED GRADIENTS Let f:R n ->• R n be a l o c a l l y l i p s c h i t z function. By Rademacher's theorem, f i s d i f f e r e n t i a b l e almost everywhere although i t need not be continuously d i f f e r e n t i a b l e anywhere. If V i s a subset of BLn, coV w i l l denote the convex h u l l of V i n H n. D e f i n i t i o n (see [3] ) The generalized gradient of a l i p s c h i t z function f at a point x e B.n,' 8 f (x) , i s defined as follows: 3 f ( x ) = co{Xel n| X = L I M Vf(x ) , x' = L ™ x }. 1 n-*o° n n-**> n (This d e f i n i t i o n has been extended to a l a r g e r c l a s s of functions -see [3] ). The f o l l o w i n g examples i l l u s t r a t e the generalized gradient of two simple f u n c t i o n s . Example Let f( x ) = | x | , then 3f ( x ) = | {1} i f x > 0, {-1} i f x < 0, [-1,1] i f x = 0. 2 Example Let f ( x ) = x s i n ( l / x ) . For x f 0 f'(x) = 2xsin(l/x) - c o s ( l / x ) . For x = 0, using basic p r i n c i p l e s , we f i n d f'(0) = 0. The generalized gradient, however, i s given by 3 f ( x ) = {f'(x)} i f x f 0, [ -1,1] i f x = 0. 2 1 The generalized gradient 3f(x) i s a closed, compact, convex, non-empty subset of R n (see [3] ). If L is a loca l l ip sch i tz constant for f about x, then i t is easy to see that for a l l A e 3f(x), If f i s continuously differentiable at x then clearly 2 9f(x) = (Vf(x)}. It may happen, as in the example f(x)=x s in ( l / x ) , that Vf(x) exists but 3f(x) f (Vf(x)}. In any case, we are always assured of the following property: Property 1 If Vf(x) exists , then Vf(x) e 3f(x). The l ip sch i tz property of f provides a very useful form of continuity for the generalized gradient: Property 2 The generalized gradient is upper semi-continuous oo (U.S.C.) , that i s ; i f {x } converges to x, and A e 3f(x ) n n - l a — n n oo for 1 < n <°°, then any l imi t point A to {A } , sat isf ies A £ 3f ( x ) . _ £ c — n n=_ Notice that {A } i s assured of having at least one l imi t n n - l & point since for large n, A_^  l i e s within the (compact) sphere of radius L where L is a local l ipschi tz constant for f about x. 22 D e f i n i t i o n (See [3] ) The generalized d i r e c t i o n a l d e r i v a t i v e of a l i p s c h i t z function  f at a point x e R n i n a (non-zero) d i r e c t i o n v e I n 5 f°(x;v), i s given by f°(x;v) = LIM SUP [(f(x+h+6v) - f(x+h) )/6 ]. h + 0 5 + 0 Like the usual one sided d i r e c t i o n a l d e r i v a t i v e , denoted f'(x;v) when i t e x i s t s , the generalized d i r e c t i o n a l d e r i v a t i v e i s a map-ping from R n X _.n to H. Notice that from the d e f i n i t i o n of f° we have the following: Property 3 f' (x;v) <_ f°(x;v) for a l l x,v i n l n . Example Let f(x) = -|x|, and consider f°(0;v). Notice that | <5v| _> |h| - |h+6v| and so |v| _> LIM SUP (-| h+6v |+| h| ) / 6. h -»• 0 6 + 0 If we l e t h =-6v then the reverse i n e q u a l i t y i s established and we conclude that f°(0;v) = |v|. This i s i n contrast to the c l a s s i c a l s i t u a t i o n , i n which f'(0;v) = — jv} . An equivalent, and often more convenient, d e f i n i t i o n of the generalized d i r e c t i o n a l d e r i v a t i v e i s given by f°(x;v) = MAX { A- v| X z 9f(x) } 23 ( see [3] ). The maximum i s attained at an extreme point i n 3f ( x ) , and since a l l extreme points are of the form LIM Vf(x ) f o r some sequence ix^} converging to x, the fo l l o w i n g property holds: Property 4 f°(x;v) = v • LIM Vf(x ) f o r some sequence {x } , converging n a n n - i n-x» to x. Suppose now that f'(x;v) e x i s t s . Then MAX {X«v|\e3f(x)} = f°(x;v) > f*(x;v) = -f'(x;-v) > -f°(x;-v) = -MAX { -A*V|A E 9 F ( 2 i ) > = MIN { X-v|X e 3f(x) } and we have the following property: Property 5 f*(x;v) e:{X-v|X e 3f(x)} = [-f°(x;-v) , f°(x;v)] . Let E be any set of zero measure. Define 3f on 5i n as E Bf^Cx) = co{XEttlrXiF; LIM Vf(x ),x -> x as n-**>, and x / E }, E n n n n-*» and define the function f° on Rn X S.n as 24 f ° ( x ; v ) = LIM SUP h 0 6 + 0 { [f (x+h+6v)-f (x+h)] ,/6 | x+h,x+h+6v i E}. As i s demonstrated in [3] , the following equalities hold: Property 6 3f (x) f ° ( x ; v ) and so also The generalized gradient and direct ional derivative extend the concept of the subdifferential of a convex function. If f i s a convex function then (i) the generalized gradient is ident ical to the sub-d i f ferent ia l Several results employing the subdifferential have been extended by the use of the generalized gradient (for an example and a br ief discussion see [4 , introduction] )• We w i l l have occasion to consider the generalized gradient of a function at an end point of the interval on which i t is defined. In this s ituation the generalized gradient w i l l be determined by l imits of sequences restr icted to the interval of def init ion of the function. If f : [ 0 , l ] -»- 51 is l i p s c h i t z , ( i i ) f ° ( x ; v ) = f ' (x ;v) ( i i i ) f(x) = {a} i f and only i f Vf(x) exists and a = Vf(x). 25 3 f ( l ) = co{a|a=LIM Vf(x ),x x e [0,1]} . n n n n-*» This corresponds to extending f symmetrically about 1, and so the properties given i n t h i s chapter w i l l hold, with the appropriate r e s t r i c t i o n s . Notice that the "one-sided" generalized gradient described here w i l l be a subset of the "standard" generalized gradient, i f the l a t t e r e x i s t s . 26 CHAPTER I I I THE GENERALIZED HAMILTON-JACOBI EQUATION R e c a l l Caratheodory's d e f i n i t i o n of equivalent problems; we w i l l generalize the concept as follows: DEFINITION One problem, or i t s integrand F, i s sa i d to be equivalent to * 2 another (with i t s integrand F ) on a set U e R i f there e x i s t s a l o c a l l y l i p s c h i t z f u n c t i o n R with i t s generalized gradient defined on U and s a t i s f y i n g F (a,b,q) = F(a,b,q) - R°(a,b;l,q) for each (a,b) e U and q e R. * If i n the above d e f i n i t i o n F i s a n i c e integrand we have, f o r (a,b) i n U and (u,v) r e s t r i c t e d to 3R(a,b), that INF [F(a,b,q) - R°(a,b;l,q)] q = - SUP [ MAX (u,v) • (l,q) - F(a,b,q) ] q (u,v) = - MAX { SUP [vq - F(a,b,q)] + u } (u,v) q = -..MAX [ H(a,b,v) + u ] (u,v) = 0 . 27 DEFINITION The relat ion MAX [ H(a,b,v) + u ] = 0 (a,b) e U (u ,v)e8R ( a ,b ) w i l l be referred to as the generalized Hamilton-Jacobi equation 2 (G.H.-J.Eq.) for R on the region U e H . Because the generalized gradient 8R i s a compact set and the Hamiltonian is convex in the third variable (hence continuous on open sets, |where i t i s f i n i t e ) , the use of a maximum, as opposed to a supremum, i s jus t i f i ed in the above def in i t ion . Since H may assume values of °° however, the G.H.-J .Eq. impl i c i t ly requires that H(a,b,v) be f i n i t e for a l l (u,v) in 3R ( a ,b ) . Notice that in analogy with the c la s s ica l case, a l oca l ly l ipschi tz function R w i l l satisfy the G.H.-J .Eq. on U i f and only i f i t establishes an equivalence between the problem and a nice problem on U. * If F , given by F*(a,b,q) = F(a,b,q) - R ° ( a , b ; l , q ) i s known only to be posit ive, then we find that R sat is f ies MAX [ H(a,b,v) + u ] <_ 0 , (u ,v)e9R ( a ,b ) which w i l l be referred to as the generalized Hamilton-Jacobi inequality  for R (G.H.-J.Ineq.) . A loca l ly l ip sch i tz function w i l l establish 28 an equivalence with a p o s i t i v e integrand on U i f and only i f i t s a t i s f i e s the G.H.-J.Ineq. on U. The following i s the c e n t r a l r e s u l t . THEOREM I NECESSARY CONDITIONS a) Let S(x,y) = SUP | F(x,y(x),y'(x) )dx yeA(x.y) J aQ where A(x,y) = { curves to (x,y) which l i e i n the domain U almost  everywhere }. If_ ( i ) S i s defined on a neighbourhood >7 C U of a point (a,b), ( i i ) S i s l o c a l l y l i p s c h i t z i n j] , and ( i i i ) F(x,y,q) i s continuous i n a l l three v a r i a b l e s , then F(a,b,q) - S°(a,b;l,q) > 0 f o r a l l (u,v) e 9S(a,b) and q E R. b) I f , i n ad d i t i o n , i s an optimal curve to (a,b) then F(a,b,q o) - S°(a,b;l,q o) = 0 for some q^ e 9y^(a). 29 Part (a) of the theorem states that inter ior to where S is defined and loca l ly l i p s c h i t z the problem w i l l be equivalent to one with a positive integrand. As we have seen, this i s the same as saying that where S i s defined and loca l ly l i p s c h i t z , i t w i l l satisfy the G.H.-J . Ineq. The value function is thus seen to be closely linked to the Hamiltonian function by the G.H.-J . Ineq. S imilar ly , part (b) of the theorem assures us that under stronger hypotheses S w i l l be a solution to the G . H . - J . E q . This relationship between the value function and the Hamiltonian function is futher considered, i n i t s c l a s s i c a l setting in [7,chapter 9] . Notice that y Q need not be defined outside the interval , [a Q ,a] , hence 9yQ(a) requires the interpretation discussed at the end of Chapter II . Proof of Theorem I In addition to the hypotheses of the theorem, assume that VS(a,b) exists . For q e 1 le t y q (x) = b + (x - a)q be the l i n e through (a,b) with slope q. Select e > 0 small enough that a < a - e and the l ine segment {(x,y (x)) I xe[a-e,a+e] } l i e s o q within TJ . Let x^ and x 2 l i e i n [a-e,a+e] with x^ < x 2 and le t y^(x) be a 6-near optimal curve from ( a 0>b o) to (x^,y^(x^)). Define 9&M e A ( x 2 > y q ( x 2 ) ) as y 6 (x) = y g (x) i f ± x < X ; L , y (x) i f x 1 < x < x 2 . 30 As a function of x, S(x,y q(x)) i s local l y l i p s c h i t z on [x^,x2] and so -^S(x,y q(x)) exists almost everywhere on [x^,x^] .. Now, I X ' [ F(x,y q,y;) - ^ S ( x , y q ( x ) ) ] dx x l j ^ F (x ,y f i ,y ' )dx - j 1 F ( x ,y f i , y ' )dx - f  2 ^S(x,y q(x))dx J ao J ao J X l > S(x 2,y (x 2)) - [S(x l Sy ( X ; L)) + 6] S(x 2,y q(x 2) - S(x 1^y q(x 1)) Since we may independently choose 6 a r b i t r a r i l y small and since the interval [x^,x 2] C[a-e,a+e] is arbitrary, we see that for almost a l l x in [a-e,a+e] , -j-S(x,y (x)) exists and Q X C[ F(x,y q(x),q) - -^S(x,y q(x)) _> 0. Let E = {xe[a-e,a+e] | -^S(x,y q(x)) does not exist, or F(x,y q(x),q) - ^ S(x,y q(x)) < 0 } , then E has zero measure. oo select a sequence {x } , i n [a-e,a+e] with x a as n -> °° n n=l n oo and {x} - n E = ^, then recalling that F i s continuous, n n=l 31 LIMSUP [ F(x n ,y (x ),q) " ^ ( ^ . Y ( x n » ] n -»• 0 0 = F(a,b,q) - LIMINF (x .y ( x n ) ) ri -> °° > 0 Denote the generalized gradient of S(x,y^(x)) as a function in x alone, as 3 S(x,y (x)), (not to be confused with 3S(x,y(x)) ). x q Now, 3 S (a,y (a)) . = co { a | a = LIM (x Q,y (X q)) , with x 4 E and x a as n °° } , n ' n and as we have seen, for each a: F(a,b,q) - a > 0 . Taking the convex h u l l of the a's preserves this property, that i s ; V a e 3 S „ ( a , y (a)), F(a,b,q) - a > 0. By property 6 of chapter II. x E q V a e 3 S(a,y (a)), F(a,b,q) - a j> 0. In part icular , S(a,y (a)) X C[ Q X l i e s in 9 S(a,y (a)) (property 1 of chapter II) and also, x y-S(a,y (a)) = VS (a,b) • ( l ,q) , so: dx q F(a,b,q) - VS(a,b)- (l ,q) _> 0 . (Equation 1) We now drop the assuption that VS(a,b) exists . Let (u ,vQ) e 3S(a,b) be such that S ° ( a , b ; l , q ) = ( u 0 , v 0 ) •(l><l) 32 and such that there is a sequence ^ a n ' ^ n ^ n - i converging to (a,b) with (u ,v ) = LIM VS(a,b). Existence of (u ,v ) is assured by ° ° o o property 4 of chapter II. For n large enough that (a ,b ) lies n n interior to 77 we have F(an,bn,q) - VS(an,bn) • (l,q) > 0 , and so, taking the limit as n •+ °°, F(a,b,q) - S°(a,b;l,q) •_> 0 , as stated in (a) of the theorem. Part II We proceed to show that the minimum value is zero and that i t is always attained if an optimal curve exists. Let Y q be an optimal curve to (a,b) and select x^ > a so that (x^,yo(x^)) is interior to the neighbourhood 77 , and such that; (i) y Q ( x i ^ exists, and (ii) | ^ I F(x,yo,y^)dx = F (x^,yQ(x^),y^(x^)) at x = x± J ao Note that (i) and (ii) hold almost everywhere along y . Now o s V ^ y ^ ) ; ! ^ ) ) LTMSUP [ S(x + h + X , y (x.) •+ h + Xy'( X l)) /, . 1 1 o 1 2 o 1 (h^.h^HO - s(X;L+ h1 , y Q(x 1) + h2) ] /x 33 > LIMSUP [ S(x + X,y (x ) + Xy'(x )) - S(x ,y (x ) ) ] / X X + 0 = LIMSUP [ S(X;L+ ^ ,y o (x 1 ) + ^ (*,_)) " S( X l + A,y o (x 1 + X)) X + 0 + S(X;L+ ^,y o(x 1+ X)) - S (x 1 , y o (x 1 ) ) ] / X Let M be the l ip schi tz constant for S in a neighbourhood of (a,b). Since YQ(:x-1+ = yQ^xi^ + ^ o ^ l ^ + o(-X^> f o r s m a 1 1 X we have: S ° ( x 1 , y o ( x 1 ) ; l , y ; ( x 1 ) ) >. LIMSUP [ o(X)-M + S(x + X,y (x + X)) - S (x ,y (x.. )) ] / X A 4- 0 h | F (x ,y 0 ,y ; ) )dx (at x = x L ) . a o Since y i s a neighbourhood of (x^,yQ(x^)), part (a) of the theorem holds here, that i s : F ( x 1 , y o ( x 1 ) , y ' ( x 1 ) ) - S ° ,y Q (x^ ;1 ,y ' (x^ ) is non-negative. Since d F ( x 1 , y o ( x 1 ) > y ; ( x 1 ) ) d x x F(x ,y o , y * )dx (at x = x ) 34 we combine this with the previous inequality to get: F ( x 1 , y o ( x 1 ) , y ' ( x 1 ) ) - S ° ( x r y Q (x .^ ; l ,y^ (x^) = 0 . (Equation 2) oo Now select a sequence {x } , with the following properties: n n=l for each n _> 1 ; (i) y ' (x ) exists , o n ^ j l F ( S ' y ° ' y ; ) dx = F(x ,y Q (x) ,y ' (x)) at x = x n , ( i i i ) (x ,y (x )) l i e s inter ior to n , n o n ' (iv) x a as n -* °°, and f i n a l l y , n oo . (v) (y'(x )} . converges (to q say) o n n=l o Conditions (i) and ( i i ) are sat is f ied almost everywhere along y^ , as mentioned ear l ie r , so conditions (i) through (iv) are easily met. Since yQ i s l i p sch i t z , y' i s bounded and any sequence satisfying (i) through (iv) w i l l have a sub-sequence satisfying (v) as wel l . For such a sequence {x } let {(u ,v ) } ° ° , be a sequence n n=l n n n=l satisfying (u ,v ) E 9S (X ,y (x )) and n n n o n for each n > 1. 35 For each "Equation 2" above w i l l hold and we may rewrite i t in the form: F ( x n , y 0 ( x n ) j y ; ( x n ) ) - ( u n , v n ) . ( l , y ; ( x n ) ) = 0. Since S is l ip sch i tz {(u ,v ) } ° ° has a convergent sub-sequence and n . n n=l i t s l imi t ( u 0 » v 0 ) l i e s in 3S(a,b) (property 2 of chapter II) . S imi lar i ly q^ , the l imi t of ^ y o ( x n ^ n - i H e s I n 9 y Q ( a ) ' Taking the l imi t as n 0 0 in the above equation we get F(a,b,q Q ) - ( U O , V q ) • ( l ,q Q ) = 0 . Now F(a ,b,q ) - MAX [(u,v) • ( l ,q )] ° (u,v)e3S(a,b) = F(a ,b ,q o ) - S ° ( a , b ; l , q o ) < F(a ,b,q d ) - (u Q ,v o ) • ( l ,q Q ) .- 0, and i n consideration of (a) in the theorem, F(a,b,q Q ) - S ° ( a , b ; l , q o ) = 0 completing the proof. 36 Notice that "equation 2" i n the proof of theorem I provides a g e n e r a l i z a t i o n to Caratheodory 1s fundamental lemma ( [ 7 , § 5 . 1 ] ) : under the hypotheses of the theorem, i f y i s optimal and y' i s continuous at a point x with (x,y(x)) within n , then conditions ( i ) and ( i i ) at the beginning of part II of the proof hold, and so according to "equation 2" ; MIN [F(x,y(x),q) - S°(x,y(x);l,q)] q = F(x,y(x),y'(x)) - S°(x,y(x),y'(x)) = 0 . I f S i s continuously d i f f e r e n t i a b l e as i s assumed by Caratheodory's lemma, then S° coincides with the c l a s s i c a l d i r e c t i o n a l d e r i v a t i v e . Although the value function w i l l not be the only s o l u t i o n to the G.H.-J.Ineq., any s o l u t i o n can be used to e s t a b l i s h a lower bound on the value f u n c t i o n , as we see i n the following lemma. LEMMA II LOWER BOUNDS ON THE VALUE FUNCTION Let F(x,y,q) be an integrand y i e l d i n g the Hamiltonian H(x,y,p). Let R(x,y) be a l o c a l l y l i p s c h i t z function defined on an open region ft and s a t i s f y i n g the generalized Hamilton-Jacobi  i n e q u a l i t y H(a,b,X 2) + X x 1 0 ( W e 8R(a>b) at each (a,b) i n ft . 37 If z i s any l i p s c h i t z curve j o i n i n g ^a^'^j} e ^ a n c * (a^jb^) £ ft with a^ < a^ and (x,z(x)) i n ft f o r almost a l l x e [a^,a ], i t follows that / z F(x,z(x),z' (x))dx >_ R(a 2,b 2) - R(a 1,b 1) a l Proof of lemma II For almost a l l x e [ a ^ j a 2 ] z ' ( x ) e x i s t s , the G.H.-J.Ineq. holds at (x,z(x)), and 4~"R(x,z(x)) e x i s t s . The equivalent integrand dx * o , F (x,y,q) = F(x,y,q) - R (x,y;l.q) i s p o s i t i v e on ft, and so by property 3, cnapter I I : F(x,z(x),z* (x)) - ^ R ( x , z ( x ) ) 0 / for almost a l l x e [a^,a 2] . Integrating, we get the desired r e s u l t : L F(x,z(x),z'(x))dx - R(a 2,b 2) + R(a^,b^) > 0 S l Under the hypotheses of lemma I I , l e t the'point (a ,b ) l i e o o i n the closure of ft; ft . Assume also that we can define R(a ,b ) o o as: R(a ,b ) = LIM' R(a,b) ° ° (a,b)->(a o,b o) 38 where LIM' ind i c a t e s the l i m i t with (a,b) r e s t r i c t e d to ft . If we l e t ft coincide with the domain of the problem; that i s , f o r (a,b) i n ft l e t A(a,b) = {curves y | y(a)=b and y l i e s i n ft almost everywhere} , then we have f a INF j F(x,y,y')dx _> R(a,b) - R(a o,b Q) . yeA(a.b)J a Q ° ° We see then, that the value f u n c t i o n i s the l a r g e s t (majorizing) l o c a l l y l i p s c h i t z s o l u t i o n to the G.H.-J.Ineq. which can be extended continuously by s e t t i n g S ( a o , b Q ) = 0 . Often however, the following question i s of more p r a c t i c a l i n t e r e s t : 'given a l o c a l l y l i p s c h i t z function R (which we suspect of being the value f u n c t i o n ) , under what conditions are we assured that i t i s i n f a c t the value function?'. A set of s u f f i c i e n t conditions i s provided by lemma III below. LEMMA I I I SUFFICIENT CONDITIONS FOR THE VALUE FUNCTION 2 Let ft e 1 • be a set of termination and l e t U , the domain of the problem, coincide with ft . If the integrand F i s continuous then a function R defined on ft U {(a ,b )} which i s o o l o c a l l y l i p s c h i t z on ft w i l l be the value function f o r the problem i f and only i f , f o r (a,b) r e s t r i c t e d to ft: 39 ( i ) R s a t i s f i e s the G.H.-J.Ineq. on ft , ( i i ) R(a ,b ) = LIM R(a,b) = 0 , and ° ° (a,b)-(a o,b o) ( i i i ) V (a,b,) eft 3 {y } . C A(a,b) such t h a t : n n=l J o LIM I F ( x , y ^ , y ^ ) d x = R(a,b) n - H » P r o o f o f lemma I I I N o t i c e t h a t s i n c e U = ft, (a ,b ) must l i e i n the c l o s u r e o f ft o o o t h e r w i s e A(a,b) = $ f o r a l l ( a , b ) . F o r n e c e s s i t y ; ( i ) f o l l o w s from p a r t (a) o f theorem I , w h i l e ( i i ) and ( i i i ) a r e consequences o f the d e f i n i t i o n o f t h e v a l u e f u n c t i o n . C o n s i d e r now the s u f f i c i e n c y o f the c o n d i t i o n s . Lemma I I and c o n d i t i o n s ( i ) and ( i i ) p r o v i d e t h a t : INF / F ( x , y , y ' ) d x _> R(a,b) - R(a ,b ) yeA(a,b) / aQ ° ° = R(a,b) , w h i l e - c o n d i t i o n ( i i i ) p r o v i d e s the r e v e r s e i n e q u a l i t y . T h i s c o m p l e t e s t h e p r o o f . i 40 Very often the ba s i c problem i s posed, not to f i n d the minimum i t s e l f , but rather, to obtain the optimal curve(s) which provide the minimum. Having imbedded the problem i n a family of problems, the theory i n v o l v i n g the solutions f o r the e n t i r e family should help solve the o r i g i n a l basic problem. In theorem II below we f i n d p o i n t -wise s u f f i c i e n c y conditions f o r opt i m a l i t y i n a basic problem. THEOREM I I SUFFICIENCY CONDITIONS FOR OPTIMALITY Let R(x,y) be 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 to the generalized 2 Hamilton-Jacobi i n e q u a l i t y on a open set f! C I . If V q i s a l i p s c h i t z curve to (a,b) l y i n g i n ft almost everywhere and i f almost everywhere f or x e [a ,aj , then y i s an optimal curve for the basic problem: f i n d F ( x , y o ( x ) , y ' ( x ) ) - -^R(x,y o(x)) = 0 INF yeA(a,b) with A(a,b) = {curves y y(a)=b and y l i e s i n ft almost everywhere} . 41 Notice that ft i s both the set of termination and the domain of the problem. In practice i t may be desired to find a solution over a domain U which properly includes the open set ft. In this case, under the hypothesis of theorem II, the curve y may be considered a o local solution. More precisely, i f y^ l i e s entirely within ft , then i s a strong loca l solution to the basic problem, and i f the domain U coincides with ft, then y^ is a global solution (see [7,§2.1]). Since theorem II does not require that R be the value function but only a solution to the G.H.-J .Ineq. , solving a basic problem need not involve solving an entire family of problems. Proof of theorem II Integrating o we get: F(x,y ,y')dx - R(a,b) + R(a ,b ) = 0 . o o o o From Lemma II , for a l l y in A(a,b), F(x,y,y ' )dx -R(a,b) + R(a o ,bQ) >_ 0 , and so y is optimal in A(a,b), completing the proof. 42 Notice that since R s a t i s f i e s the G.H.-J.Ineq., F( x , y o ( x ) , y ' ( x ) ) - R°(x,y o(x);1,y'(x)) > 0 wherever y' e x i s t s and (x,y o ( x ) ) e ft. Since ^ R ( x , y o ( x ) ) < R°(x,y o(x);l,y' (x)) where y^(x) e x i s t s , we have 0 = F ( x , y o ( x ) , y I ( x ) ) . - ^ R ( x , y Q ( x ) ) >_ F(x , y o ( x ) , y ' (x)) - R° (x,y Q (x) ; 1 ,y' (x) ) > 0, and so ^ R ( x , y o ( x ) ) = R°(x,y o(x);l,y'(x)) i n ft wherever y' e x i s t s , o 43 CHAPTER IV EXAMPLES The generalized gradient of the value function can be considered as a closed compact convex region in the x-y plane. If i t i s rotated 9 0 ° clockwise about the or ig in (that i s , {(x,y)} {(-^y,x)} ) then the G.H.-J .Eq. can be expressed by saying that 3S(a,b) w i l l l i e above the graph of y = H(a,b,x) and w i l l touch the graph at one or more points. EXAMPLE I Let the integrand F be the following: F(x,y,q) q 2 - l i f |q| _> 1 , 0 i f |q| < 1 . Let the set of termination be ft = {(x,y) | x > 0 }, and the domain 2 be 1 . Let (a o ,bQ) = (0,0) then for each (a,b) e ft the set of admissible curves i s A(a,b) = {l ipschitz curves from (0,0) to (a,b) } . For each (a,b) in ft consider the straight l ine curve y(x) = bx/a, which l i e s in A(a,b) . These satisfy the necessary condition of the Euler-Lagrange d i f ferent ia l inclusion for optimality - see [4,theorem 2.4] Define R, a l ip sch i tz function on ft as: R(a,b) = f F(x,bx/a,b/a)dx J 0 (b 2 - a 2 ) /a i f lb/a| > 1 , i f |b/a| <_ 1 44 The Hamiltonian for the integrand F is : H(a,b,p) = SUP q p q - q + 1 i f | q | _ > l » pq i f Iq l < 1 p /4 + 1 i f |p| _> 2 , |p| i f |p| < 2 . H(a,b,p) i s independent of (a,b) e ft, and a sketch of H as a function of p is given in figure I below. The generalized gradient of R(a,b) i s given by: 3R(a,b) = | {(-(a2+ b 2 ) / a 2 , 2b/a )} i f |b/a| > 1 , {(0,0)} i f |b/a| < 1 , {(-X,X) \ 0 < X < 2 } i f b/a = 1 {(-X,-X) | 0 £ X < 2 } i f b/a = -1 This is represented in figure II below. Notice that when 3R(a,b) is rotated clockwise i t always l i e s on the graph of H(a,b,x) for any (a,b) in ft, and so R satisf ies the G.H.-J .Eq. on ft. We can now veri fy each of the hypotheses of theorem II (i) y(x) = bx/a l i e s i n ft almost everywhere, ( i i ) F (x ,y ,y ' ) = •— R ^ . y ^ ' ) , ( i i i ) R is l ipschi tz on ft, and (iv) R satisf ies the G.H.-J.Ineq. on ft; hence each of the curves of the form y(x) = bx/a are optimal curves to (a,b), and R coincides with the value function. These curves, however, 45 DIAGRAM II 3S(a,b) for Some Values of a/b, Example I 46 are not unique optimal s o l u t i o n s since, f o r 0 < b / a < l , i t i s easy to construct others such as y(x) = MIN{x,b}. EXAMPLE II 2 2 Consider the integrand F(x,y,q) - q - y . From c l a s s i c a l theory we f i n d that optimal curves, i f they e x i s t , should be of the form y(x) = "k*.sin(x), k e IL (see [7, §2.3] ). A l l such extremals pass through the point (TT,0) which i s a conjugate point (again, see [7,§3.6] ). Let ft, the set of termination, be given by: ft = {( X,k-sin'x) | 0 < x < 2TT, x f IT, |k| < 1 } . Notice that ft i s open and disconnected. Let the domain be ft and l e t (a ,b ) = (0,0) then the set of admissible curves to (a,b) e ft w i l l be: o o A(a,b) = { l i p s c h i t z curves from (0,0) to (a,b) l y i n g i n ft a.e.} For each (a,b) i n ft there i s a unique y e A(a,b) of the form k * s i n x: y , (x) = k , s i n x with k , = b / s i n a. Define R on ft as: ab ab ab R(a,b) = J F ( x , y a b , y ; b ) d x J 0 ,2 = b cot a . Let R(0,0) ~ 0; then R i s continuous on ft U {(0,0) > and R i s seen to be l i p s c h i t z on ft,with ||VR(a,b)|| = | | ( - b 2 / s i n 2 a , 2bcot a)| | < 5 . 47 The Hamiltonian for the problem is found to be H(a,b,p) = p2/4 + b 2 and i t easily seen that R satisf ies the G.H.-J .Eq. on ft. By construction ^ R ( x , y a b ( x ) , y ^ b ( x ) ) = F ( x , y a b (x) ,y ^ (x) ) so by corollary II y a b i s optimal in A(a,b) and hence R is the value function on ft. In a similar fashion, i f we let ft = {(x,ksin x) | 0 < x < 2u, X f ir, j k| < m} , and A (a,b) - { l ipschi tz curves from (0,0) to (a,b) within ft a.e. }, m m then on ft , y , w i l l be optimal in A (a,b). For a > T T , any l ip sch i tz m ab m curve from (0,0) to (a,b) which passes through (IT,0) must l i e within ft almost everywhere for some m, hence our curves y , are optimal m ab over a l l l ip sch i tz curves y(x) with y(0) = 0, y('ir) = 0, and y(a) = b. Notice that i f the point (TT,0) was included i n the sets ft or ft , m then our curves y could no longer be optimal for a > TT by Jacobi's ab necessary condition (see [7^3.6] ) EXAMPLE III Consider the smooth integrand F(x,y,q) = y ( l + q V / 2 . Let the set of termination be ft = {(x,y)| x > 0, y > 0}, let the 48 domain coincide with ft, and le t (a Q ,b o ) - (0,1) then the set of admissible curves to (a,b) e ft i s given by A(a,b) = {lipschitz curves from (0,1) to (a,b) lying within ft a.e.}. This is a very old and often cited problem from the c la s s ica l theory of the calculus of variat ions . It corresponds to minimizing the area of revolution of a posit ive function. Physical ly, the problem seeks the shape of a soap bubble spanning two concentric hoops. Experience indicates that as the two hoops are moved away from one another the bubble eventually breaks. We w i l l see that this happens, not because of a i r currents or insufficient soap or any other accident but because eventually, l oca l ly optimal curves for the problem f a i l to exist , making the soap f i lm unstable. Of particular interest i s the fact that the value function f a i l s to be differentiable despite the fact that the integrand sat is f ies the c lass ical requirements of being twice continuously differentiable and convex. In regions where there are no optimal curves we w i l l be able to establish the value function, while with the same tools, in regions which have optimal curves we w i l l find them and the value function. Where strong loca l solutions are found, a well defined locale within which they are optimal i s also found. The solutions to the Euler-Lagrange equation ( 2,§273 ) are X—C of the form y(x) = d-cosh(-j-) . To each point (a,b) e ft there are at most two curves of this form in A(a,b) (see diagram III below). x—c The ensemble of curves of the form y(x) = d*cosh(——) with y(x) e A(a,b) 49 for some (a,b) e ft forms an envelope c u r v e , ' E ' , as in diagram I I I . If we truncate each of the curves of the ensemble at the point where i t touches the envelope (see [7,§A3.13]), there remains exactly one member of the ensemble lying in A(a,b) for each (a,b) e ft lying above the envelope ' E ' . The constants c and d are smooth functions of the coordinates (a,b) above ' E ' ; y a b ( x ) - d ( a , b ) . c o s h ( ^ ^ . ) Define the loca l ly l ip sch i tz function on the region above the envelope ' E ' as follows: R l ( a , b ) = • F ( x >y a b >y ' a b > d x 2.\ da + d*sinh(—-—) -b + d«s inh(—) ] , d d with c - c(a,b) and d - d(a,b). The Hamiltonian for the problem is given by: H(x,y,q) = SUP [ pq - F(x,y,q) ] q " ( y 2 - P 2 ) 5 i f y 1 0 and |p| _> y + 0 0 otherwise . The gradient of R^ is found to be V R ; L(a,b) = ( d , i j - S l . (b 2 -d 2 )^ ) , 50 (again with c = c(a,b) and d = d(a,b) ) which satisf ies the G.H.-J.Eq. on the region where is defined. The hypotheses of the lemma III are sat is f ied on the region above the curve ' E ' , and so the curves y ^ a r e optimal in A'(a,b) E { y e A(a,b) | y l i e s above ' E ' almost everywhere } . For the or iginal problem however, we are assured only that the y ab are strong local solutions. In search of the value function we w i l l require a function defined on a l l of ft . CO For each (a,b) E ft consider the sequence of curves {y } _.. n n-1 defined as follows: 2 y n(x) = l-nx i f 0 < x < (n-l) /n 1/n i f (n - l ) /n 2 <_ x < a- (nb- l ) /n 2 2 b-n(a-x) i f a-(nb-l)/n < x < a . For suff iciently large n, y^ i s ah element of A(a,b) and we find that LIM I F(x,y ,y')dx = (b2+ l)/2 I „ n n n-*» 2 Set R2(a,b) equal to this value, (b + l)/2, for each (a,b) in ft. As computation ver i f i e s ([1]) R^ = R^ along a curve 'G ' which l i e s above ' E ' (see diagram III ) . Above the curve ' G ' we have R^ < R^, while between 'G' and *E ' we have R^ > R,,. Notice that R 2 is smooth and VR 2(a,b) = (0,b) on ft, hence R 2 sat is f ies the G.H.-J .Eq. Notice however, that since R „ ( a ,b ) f 0, R„ cannot be the value 2 o o 2 52 function. By combining and R,, we w i l l attempt to s a t i s f y a l l the hypotheses of lemma III . Define R on ft as follows: R(a,b) = MIN {R ,R } on the region above ' E ' R 2 elsewhere on ft . The function R i s l ip sch i tz on ft, but f a i l s to be differentiable along the curve ' G ' . The generalized gradient of R is as follows: 3R(a,b) = |{VR (a,b)} i f (a,b) is above 'G* {VR2(a,b)} i f (a,b) is below 'G' Jco{VR 1(a,b),VR 2(a,b)} i f (a,b) is on ' G * . Since; (i) R sat is f ies the G.H.-J .Eq. on ft,' ( i i ) R(0,1) = 0, and ( i i i ) R is constructed from curves or l imits of sequences of curves in A(a,b) , we conclude by lemma III that R is the value function. We have seen that to the right of ' E ' , that is for large hoop separation, no f i lm of minimal surface area exists , hence no bubble is expected to be observed. The minimum i s obtained as the l imit of a sequence which, loosely speaking, tends toward a s i tuation in which each of the hoops has a f lat film within i t s circumference. This is known as the Goldsmidt solution and i t provides the minimal surface area in the parametrized form of the problem. Between the curves ' G ' and ' E ' the catenaries defining exist , but are not optimal. In c las s ica l theory they are cal led strong loca l solutions. It can easi ly be seen that they are optimal in the smaller set of admissible functions: 53 A'(a,b) = { y £ A(a,b) | y l i e s above ' E ' } . We would expect that soap bubbles may exist for (a,b) between ' E ' and ' G ' , but that they would not be stable except under suff ic ient ly small perturbations. EXAMPLE IV = P-|y|+ Consider the problem with the integrand F(x,y,q) = e domain U = { (x,y) e 1 | x > 0 } and set of termination ft = U. Let the i n i t i a l point (a ,b ) be the or ig in . The integrand i s o o neither convex nor smooth, so the c la s s ica l Hamiltonian is not defined anywhere. The generalized Hamiltonian, however, is defined as follows: H(x,y,q) -e - y i f p = 0 otherwise For a point (a,b) £ ft let {y } n be the sequence of n n=l admissible curves to (a,b) given by y n(x) = nx f- n n - l for 0 < x < - z — • a — — 2 n / -. \ . b _ n—1 ( n - l ) « b -nx for ——«a < x < a a zn — — we find that: LIM / F(x,y y' J o n n )dx Let R(a,b) = 0 for a l l (a,b) in ftU{(0,0)}, then R sat is f ies the - I b l G.H.-J.Ineq. on ft since H(a,b,0) < 0. Since R sat is f ies 54 a l l the hypotheses of lemma III , R is the value function on Q. Notice however, that R f a i l s to satisfy the G.H-J.Eq. Since existence of optimal curves would guarantee that R satisf ies the G.H.- J .Eq . (by theorem I part (b)) we must conclude that no optimal curves exist . 55 REFERENCES G. A . B l i s s , 'Lectures on the Calculus of Variat ions ' , Chicago 111., University of Chicago Press, 1946. C. Caratheodory, 'Calculus of Variations and Part ia l Di f ferent ia l Equations of the 1st Order' , (1937), translated by R.B.Dean, Holden-Day Inc. , San Franciso Ca. , 1967. Frank H. Clarke, 'Generalized Gradients and Applicat ions ' , Trans. Amer. Math. Soc. 205 (1975) pp247-262. Frank H. Clarke, 'Euler Lagrange Dif ferent ia l Inclusion' , J .Di f ferent ia l Equations V o l . 19 (1975) pp80-90. I.M.Gel'Fand and S.V.Fomin, 'Calculus of Variat ions ' , translated by R.A.Silverman, Englewood C l i f f s N . J . , Prentice-Hall 1963. H. Goldstein, 'Class ica l Mechanics', Addison Wesley, Reading Mass.,1950. Hans Sagan, 'Introduction to the Calculus of Variat ions ' , N.Y.,McGraw-Hill 1969. D. R.Snow, 'A Sufficiency Technique in the Calculus of Variations i Using Caratheodory's Equivalent Problems Approach', J . of Math. Analysis and Applications V o l . 51 (1975) ppl29-140. 

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