STABILITY I N T H E L A R G E O F AUTONOMOUS SYSTEMS OF TWO D I F F E R E N T I A L E Q U A T I O N S by I Z H A R - U L HAQ M U F T I A THESIS S U B M I T T E D I N P A R T I A L FULFJLMEffl OF THE E E Q U I E i M E N T S FOR THE DEGREE OF D O C T O R OF P H I L O S O P H Y in the Department of MATHEMATICS We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o the standard required from candidates for the degree of Doctor of P h i l o s o p h y . Members of the Department of Mathematics T H E U N I V E R S I T Y O F April, B R I T I S H I960. C O L U M B I A In 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 of the r e q u i r e m e n t s f o r an advanced degree at the University o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t freely study. a v a i l a b l e f o r r e f e r e n c e and 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 f o r s c h o l a r l y purposes may I further c o p y i n g of t h i s be g r a n t e d by t h e Head of Department or by h i s r e p r e s e n t a t i v e s . g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n Jy)A.lJLfcWvA.^/'X The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. Date my I t i s understood t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r Department of thesis financial permission. (Elu* Miiiuersiiy uf British OluUunbia Faculty of Graduate Studies PROGRAMME OF THE FINAL O R A L E X A M I N A T I O N FOR T H E D E G R E E OF D O C T O R OF PHILOSOPHY G R A D U A T E STUDIES Field of Study: Applied of Mathematics Functions of a Complex Variable R. R. Christian] Theory of Functions of a Real Variable R. A . Restrepo Differential Geometry Linear Analysis and Group Representation D . Derry H . F. Davis Elementary Quantum Mechanics Spectroscopy Quantum Theory of Radiation B.Sc. (Hons.) University of Sind, Pakistan 1951 M.Sc. University of Karachi, Pakistan 1953 IN ROOM FRIDAY, Other Studies: . Statistical Theory of Matter IZHARUL H A Q MUFIT W . Opechowski F . A . Kaempffer APRIL BUCHANAN 8, COMMITTEE G . M . Volkoff A . M . Crooker 255, DEAN E. D. M. R. 1960 BUILDING AT 3:00 P. M . IN CHARGE G . M . S H R U M : Chairman LEIMAN1S DERRY MARCUS A. RESTREPO W. G. R. F. External Examiner: P R O F . J. P. S. M. E. A. TAYLOR VOLKOFF BURGESS KAEMPFFER LASALLE University of Notre Dame, Indiana S T A B I L I T Y IN T H E L A R G E O F A U T O N O M O U S S Y S T E M S OF T W O DIFFERENTIAL E Q U A T I O N S We prove a similar theorem without the requirement of (5). W e also discuss the stability i n the large of the systems ABSTRACT T h e object of this dissertation is to discuss the stability i n the large of the t r i v i a l solution for systems of two differential equations using qualitative methods (of course in c o m b i n a t i o n with the construction of Lyapunov function). T h e right hand sides of these systems do not contain the t i m e t e x p l i c i t l y . dx = dt ax + L (y) dx = f dt ( x ) = 1 First of a l l we discuss the system of'the type =ip-), ^=F(*,y), * = c x - dy, cjto (1) These equations occur i n automatic regulation. Using qualitative methods we determine sufficient conditions in order that the t r i v i a l solution of system (1) be asymptotically stable i n the large. In this connection we note that a theorem proved by Ershov (Prikl. M a t . M e h . 18(1954), 381-383) is wrong. W e then solve the p r o b l e m of A i z e r m a n for the systems of two equations, n a m e l y , for the systems | £ = f(x).+ ay.,. dy_..=-bx.+ cy , dy. dt 1 f (y). , f = dy. dt c = ( x ) 2 b x + c W + c y W e consider again the system of the type (1) under assumptions as indicated by Ershov (Prikl. M a t . M e h . 17(1953), 61-72) who has discussed various cases where the stability in the large holds. N o t agreeing fully with the proofs of these theorems we give our own proofs. F i n a l l y we discuss the stability in the large of the systems : dx = dt h « ( y ) x+ ay d i = xh„ (y) + ay dt. 1 . (2) , dy_ = h ( x ) dt , & 1 2 dt x + by = bx + h M T 9 (6) 2 and 4^=ax+f(y). dt ' .$!=bx+xy dt v / / (3) In the case of'system (2) we g i v e ^ a n e w proof of a theorem which asserts that if c +. ab i-.'o,. then the. t r i v i a l solution is asymptot i c a l l y stable i n the large under the generalized Hurwitz conditions. T h e theorem was.first proved by Erugin. For system (3) M a l k i n showed that the trivial? solution is asymptotically stable i n the large under the conditions: 2 a + c < o, (acy - bf (y)) I y ^ (acy - b f ( y ) ) dy —> o for y jt'b and + oo as lyl-> (4) + oo (5) under, suitable assumptions. A s a sample case we prove that if ab > o, then the t r i v i a l solution of system (6) is-asymptotically stable i n the large under conditions: h (y) + . r. h (x) < 2. o, K< (y) h , (x) - - ab ~? o for x jfc o,' y # o. ABSTRACT The stability two in i n object the of large this of the d i f f e r e n t i a l equations combination with The right the hand s i d e s of dissertation is trivial to discuss solution for systems u s i n g q u a l i t a t i v e methods construction of these systems Lyapunov' do n o t the (of of course function). contain the time t explicitly. First of a l l we d i s c u s s (Sec. the 2.) system of the type = These qualitative that in the the fW, equations methods t r i v i a l occur s o l u t i o n of In this b y E r s h o v [7] is wrong (Sec. the for the i n we d e t e r m i n e large. Aizerman ,. = of - system 2). (l) be regulation. Using conditions i n asymptotically we n o t e that We t h e n solve two e q u a t i o n s (1) dy automatic sufficient connection systems cx a theorem (Sec. the 3), = f(x) + problem and 1 bx + ay cy of namely,for (2) dt stable proved systems dx dt order- i i i g « the case f(y) + (3) If In ax = bx of + cy system we g i v e (2) a new p r o o f of a 2 theorem which is asserts that asymptotically stable Hurwitz Por conditions. system totically (acy - stable bf(y)) y i n > c + i n the The (3) M a l k i n if ab large the large o for y * o first the under o, then under the theorem was showed t h a t * t r i v i a l solution generalized proved by Erugin t r i v i a l the the solution is conditions a + [8], asymp- c < o, and y j We p r o v e a (acy - bf(y)) d y — + as o s i m i l a r theorem without y j (acy - bf(y)) the dy—* + « |y}_» + ~ requirement as of | y | - * + 00 o We a l s o discuss (Sec. the 4) s t a b i l i t y i n the large of the systems but under § - .x^W dx = f ^ x ) f (y) 2 (Sec. assumptions as various holds. Not agreeing our + We c o n s i d e r cussed give , cases own p r o o f s . ,f£ again - f (x) c y + 2 = bx the + the with Finally cy system of indicated by Ershov where fully 5) f stability the of we d i s c u s s (Sec. type who h a s asymptotic proofs the these 6 i n (l) dis- the large theorems a n d 7) the we . iv s t a b i l i t y i n the large of the systems |j || = ^ (y) x + ay , f| = x h / y ) + ay > ft = = h ( x ) x + by 2 b x + h 2^ ^ x y under s u i t a b l e assumptions. A s a sample case we prove that i f ab > o,then the t r i v i a l solution of system (4) i s asymptotically stable i n the large under conditions h^y) + h (x) 2 < o , h.jCy) h ( x ) - ab > 2 o,for x * 0, y * o V TABLE OF CONTENTS Page Introduction 1 1. Some basic theorems on s t a b i l i t y 4 2. S t a b i l i t y i n the large of the system f 3. - * * , y ) , g . (W 14 The problem of Axzerman f o r systems of two equations 4. 25 A generalization of the problem of A i z e r man f o r two equations 5. , 38 The s t a b i l i t y i n the large of |f = P(x,y), f£ = f(«0 using quali- t a t i v e methods only 6. The s t a b i l i t y i n the large of H 7. = ^ (y) x + ay, |f = ^(x) x + b y 55 The s t a b i l i t y i n the large of |f 8. 42 = Remarks Bibliography x h ^ y ) + ay, |f = bx + h ( x ) y 2 67 69 .70 vi ACKNOWLEDGEMENTS The author wishes to acknowledge h i s indebtedness to Dr. E. Leimanis f o r suggesting the topic of this t h e s i s , and f o r encouragement and advice received throughout the preparation of this t h e s i s . He also wishes to express his thanks to the National Research Council of Canada whose f i n a n c i a l assistance has made t h i s study possible. 1 INTRODUCTION The using i n v e s t i g a t i o n of geometrical differential or ( i . e . , on a plane) and continued by many a u t h o r s 1950 ([8],[9] tative quent nature use of for the this ), lem the is times large of theorem the i n the for a was started last system of i n large. by We h a v e of the thesis is to study the by constructing of Lyapunov f u n c t i o n s . It lem of these s t a b i l i t y by particular the sense and i n the or general methods s h o u l d be of Section of L y a p u n o v and a s y m p t o t i c on the some- we r e v i e w t h e and g i v e and sometimes by construction of that the In large Lyapunov f u n c t i o n s , and the noted criteria i n have concepts Towards the function for equation of the end of the the to of stabilities stability construction of Lyapunov functions methods. solving differential for order' the find the probeither equation. stability i n in the small i n the large and on s e c t i o n we c o n s t r u c t second stability prob- m e t h o d s we d o n o t solutions 1, fre- This q u a l i t a t i v e methods qualitative quali- our work. s o l v e d sometimes of In of made two d i f f e r e n t i a l e q u a t i o n s . basis Poincare', theorem of on the two eighty years. formulated a general i n the large systems combination based during stability The m a i n p u r p o s e in curves q u a l i t a t i v e methods equations N.P. Erugin integral qualitative a Lyapunov 2 and g i v e sufficient large the t r i v i a l s o l u t i o n of In Section 2, we d i s c u s s system of differential the of following dx conditions which = dt of q u a l i t a t i v e methods guarantee In for the Axzerman is stability the i n the equation. stability i n the large of equations , i n = where 3, two cx combination with certain s t a b i l i t y i n the Section Section 2 but i n the In large of above stability we d i s c u s s - dy, the construction sufficient conditions large. the famous problem of Aizerman equations. 4, a sort of generalization of problem of discussed. In Section the Section systems In of f(<y) Lyapunov f u n c t i o n and o b t a i n which the the P(x,y) = using ensure 5, we d i s c u s s under d i f f e r e n t large Sections using again the system of assumptions and e s t a b l i s h q u a l i t a t i v e methods 6 a n d 7, we d i s c u s s the of dx dt dt h (y) x + ay h (x) x + ay and dx dt dt xhi (y) bx + + ay hu(x) equations y the only. stability i n the 3 mostly by q u a l i t a t i v e methods. It may be noted that Gu,Cao-hao f l 6 ] has considered a s i m i l a r problem. He has discussed the s t a b i l i t y of |f I ^(y) = - - + * by constructing a Lyapunov function. ,( ) y 'W 4 1. Some Basic Theorems On S t a b i l i t y . Let us consider a system of d i f f e r e n t i a l equations dx ± X^ (x^, X g , ...,x ,t) = n (l.l) of the perturbed motion. I t i s assumed that X ( o , o, t) = i o (i = 1 , ...,n) and the right hand sides X^ of ( l . l ) are continuous functions with respect to a l l t h e i r arguments and s a t i s f y the c o n d i t i o n of uniqueness of solutions of the system ( l . l ) i n the region - «o £ X^ 4+oo - , If we denote the t o t a l i t i e s (x^, x , 2 t > O (1.2) x ) and(X^, ..., X ) by n n x and X(x, t ) respectively, each being (n x 1)matrix, then the system ( l . l ) i s written i n the form x = X(x,t) (1.3) Since i t i s assumed that X(o,t) = o, equation (1.3) admits the t r i v i a l solution x(t) = o. The motion corresponding to t h i s solution i s c a l l e d unperturbed motion and motions corresponding to a l l other solutions are known as perturbed motions. D e f i n i t i o n 1. The t r i v i a l solution x ( t ) = o i s c a l l e d stable i n the sense of Lyapunov i f , given a small f > o, there e x i s t s a % ( € >t ) such that, f o r a l l perturbed motions x ( t ) f o r which Q x(t )| Q t >y t ^ % holds, the i n e q u a l i t y | x ( t ) | <. € i s s a t i s f i e d f o r >, o. o D e f i n i t i o n 2. I f the t r i v i a l solution i s stable i n the above sense and every perturbed motion s u f f i c i e n t l y close to i t i s such 5 that I- . x(t) = of 3 1 0 o ix(t)| is then we s a y stable that the i n the t r i v i a l small or in t oo. , solution the point be If 3« (X , however t q above us sense of near x for a l l t for the same v a l u e s p l i c i t l y Erugin t stated = of the saying as >/ in o and the the of t the trivial in the for that satisfies (i). definition that solutions as motion is of t it is of (l.l) Examples can be they not are example, i n equations a l l t defined given defined the (lj) > are always solutions case are o perties in the of to where of t the is >/ i n the are understood. right the hand o.This (1.2). region bounded even can. happen, right hand sides a l l and t x ex- ° solutions >, t not but of are i f for differential not bounded ^12]). (Erugin to the Lyapunov. This the sense a l l defined for We n o w t u r n according for the motion implies o when t h e and continuous where stable [x(t)| < £ property is the defined inequality it of is o sides what said any p e r t u r b e d first though t solution (i) the The boundedness a l l matter implications two p r o p e r t i e s (ii) no large. detail that —» unperturbed the i n more o possesses fll]showed existence By in o L y a p u n o v , we u n d e r s t a n d started >/ then stable discuss definitions. the |x(t)|—> ) may b e , asymptotically Let for sense Lyapunov. the in o, asymptotically Definition to = solutions of Lyapunov and d e f i n i t i o n of concept system the asymptotic includes ( l . l ) . other is in One i s itself that i^ |x(t)| m stability two of = pro- stability o. 6 There the in are cases second the t i a l where property but sense of equations has types [23], I.G. of In t r i v i a l the s o l v i n g the motion determined. are stability tain is general made properties. we c a l l , t h e and h a v i n g in domain. domain (the i f it = $ The 5. is the depend divided the second above denote , t group any scalar o, the function V(x,t) values also of in is others. stability two of either the perturbed of satisfying second = sign of of of groups. was done b y called the. same Massera problem function V(o,t) differen- i n which derivatives where of J.L. the equations classification partial >, by into methods stable literature. and to be stability, the considered possess [l8]. on a f u n c t i o n V ( x , t ) cer- Lyapunov methods. x,t, the contin- f i r s t order o. semi definite i n that i n domain allowed). A f u n c t i o n W(x) definite o. % assumes zero Definition W(o) In continuous \x\ 4. value positive system found i n the of ( l . l ) uniform relating are a l l solutions As the V(x,t) uous Definition a two methods tkc L y a p u n o v ' * f i r s t and Let a to been methods include or of e.g. also problems the particular class N . N . K r a s o v s k i i [20], [22], g r o u p we system N.N. KrasovskiJ are have the s o l u t i o n may n o t stabilities, stability solution f i r s t of of t r i v i a l given by stabilities Malkin For the been types uniform asymptotic These solutions L y a p u n o v . One s u c h Other the the independent a domain i f W(x) > of 0 for t is a l l said x * to o be and a 7 Definition a domain, tion W(x) We s h a l l 6. if there exists such that Definition upper bound i f given jV(x,t) | ^ for such that V(x,t) is €• said to which with is either-a or is o, N, t i n the to of exists \x\ o, positive definite there func- definition. an i n f i n i t e l y a % exists > o small such a number N |v(x,t)| o follows that > M, > o then large. i n view of function i d e n t i c a l l y equal continuous i n 8 • £ function V(x,t), time definite domain of V admits > >, is definite there infinitely respect semi that every M ] be W(x) o whenever A definite 9. of for >, > >, V(x,t) a positive say ( t Jx for Definition V(x,t) If 8. that V(x,t) We s h a l l 7. Definition say to of a the the total derivative perturbed sign opposite zero, is called equations to that of a Lyapunov function. Lyapunov proved the asymptotic If motions exists there derivative, for the and a d m i t t i n g of It may b e example The large can asymptotic differential be to is noted this an on that effect proved i n the i n the of perturbed possessing a definite i n f i n i t e l y s m a l l upper bound, asymptotically this has f o l l o w i n g theorem stability equations a Lyapunov f u n c t i o n , unperturbed motion simple result stability. T h e o r e m 1.1. the following classical stable. theorem been is sense of not reversible. given by J . L . an a s y m p t o t i c same w a y a s is then Massera stability i n proved a theorem Lyapunov by J . L . A [23}. the on Massera [23]. 8 T h e o r e m 1.2. nite there function V(x,t) bound tive in If andwhich definite, an i n f i n i t e l y large possesses i s such that positive an i n f i n i t e l y small its.total time derivative defi- upper i s nega- the solution x = o i s asymptotically stable the large. far inversion of this i n i t s quite has been generality. In this differential Section = the right X hand the variables i i ^ x 1 ' ••*» n^ x sides ...,o) corresponding 1.2.1. definite then particular cases this the following system of ^ = 1 » ' " » n (1-2.1) ) are continuously differentiable functions i n t h e r e g i o n - «*> < , x. < + <*> Furthermore X (o,o, The t h e o r e m we c o n s i d e r x , , . . . , x 2,...,n. Theorem O n l y i n some proved so i dt~ = 1, has not been equations dx where theorem done. 1.2. i which then The of exists If there = o ( i = t o T h e o r e m 1.1 exists 1,2,...,n). i s the following f o r the system f u n c t i o n V ( x i , . . .TQ f o r w h i c h i s (l.2.l) negative the solution x = o i s asymptotically stable a positive definite, i n the sense of Lyapunov. This Theorem theorem 1.2.2. totically V-function If stable exists i s reversible the trivial according such that a n d we h a v e s o l u t i o n of system to Lyapunov, then 2j-g i s n e g a t i v e (l.2.l) a positive definite. i s asympdefinite 9 The terize of the above two asymptotic theorems stability show t h a t of the zero V-functions solution i n charac- the sense 1.2.1 is Lyapunov. We g i v e applied the to the small. a simple problems Consider example to concerning the the = y " method real cannot be - = roots parts of of the the applied. Let V(x,y) Clearly, by this virtue function (l.2.2) of is is roots us = Note lar o of obviously negative system (l.2.2) that we d i d n o t here solutions of the is zeros. y 2 Lyapunov* first 4 f o l l o w i n g as definite. 2 1+ ^ V-function Its time derivative L r) + definite. in Hence are 2 Hence asymptotically system approximation by ~ have first the x + o ( = of are take = given y system positive at is i n (1.2.2) " X dV which stability 3 dt i i . i . e . , the asymptotic X dy characteristic Theorem ho?ir 3 dx the show system dt x the to find order the stable either to t r i v i a l i n the general decide the or solution small. particu- stability prob- lem. guarantee enough the We n o w s h o w b y an example the stability for trivial the asymptotic establishment solution. Consider of that in the the the the V-functions small are stability system of in two not the which good large equations of 10" If where * (o) . at * y - *(*) = o, x * (x) > o = -•(*) ( 1 * 2 3 ) f o r x * o. Following Malkin [2l] the V-function f o r t h i s system can be taken 2 as V(x,y) = x + J *(x) dx o I t s t o t a l time d e r i v a t i v e i n view of (1.2.3) i s given by f -gy - -<*«> 2 C l e a r l y , V(x,y) i s p o s i t i v e d e f i n i t e and ilZ dt = < o f o r x $ o and o f o r x = o. I t can be shown that i n t h i s case asymptotic dV s t a b i l i t y i n the small holds even i f -gfc i s not negative d e f i n i t e , x Now i f J * (x) dx —f-* + as |xl—» + , then i t i s possible o 0 0 0 0 ( P l i s s [25]) that the s t a b i l i t y i n the large may not hold,i.e., we can show that there exist t r a j e c t o r i e s going to i n f i n i t y f o r t —» + 00 The above example shows that i t becomes necessary to put an extra condition on the V-functions i n order to r e a l i z e asympt o t i c s t a b i l i t y i n the large. The V-function should be such that V(x^ , X 2 , ••• •;. > ) x n ing ^ C, C > o, defines a bounded region contain- the o r i g i n f o r a l l C. Because then we can be sure of the s o l u - tions being bounded and defined f o r a l l t t >, o, no matter o what the i n i t i a l point may be. Our purpose i s served i f we pose on the V-function an a d d i t i o n a l requirement of being iminfinite- l y large, since i t i s known (Erugin (14] ) that V(x) possessing the property V(x) > o f o r x * o and V(o) = o does not define the 11 region V(x) $ C, C The due > f o l l o w i n g theorems to E . A . Barbashin 1.2.3. Theorem inite o which i s If of t r i v i a l there trary the perturbed solution i n i t i a l If the tiable i n f i n i t e l y large definite solutions It of Wintner the interval that c a n be are positive def- dV d e r i v a t i v e -g^- o f w h i c h b y is negative definite, asymptotically that t there continued stable then for the arbi- immediate be differen- time - provided that °° 4 < t o. [10], i n v e r t i n g a o f type possesses (l.2.l) continuable has been continuation i s not to the i n t e r v a l solutions asymptotically f u n c t i o n V"(x) h a v i n g respect system . I t = o is a continuously out by E r u g i n a l l i t s 1.2.4 x(t) definite to not every of exists with < + ° ° i n Theorem solution positive was p o i n t e d - «* < An is derivative the requirement t-axis equations then [30], t h a t property i n the large an i n f i n i t e l y large time t r i v i a l i n the large, a l l exists (l.2.l) stable negative on s t a b i l i t y disturbances. 1.2.4. Theorem of bounded. a n d N . N . K r a s o v s k i i [2]. function V ( x ) , the total virtue always of shown on the ([4], solutions on the theorem whole [3 !) 1 negative essential. generalization of Theorem 1.2.3 is the following, Theorem inite 1.2.5. function Let there exist V ( x ) and a s e t 4r dt < © M outside an i n f i n i t e l y such M-iSL ' d t large that £ o on M positive def- 12 Let.the set M possess the property that an arbitrary i n t e r s e c t i o n of the sets V = C (g * o) and M does not contain the p o s i t i v e h a l f t r a j e c t o r y of the system ( l . 2 . l ) then the t r i v i a l s o l u t i o n x = o of system (1.2.l) i s asymptotically stable f o r a r b i t r a r y initial disturbances. As an example of Theorem 1.2.5 tial we consider the differen- equation dt^ + * (dx) g(x) + cit f (x) (§&) = o . dt (1.2.4) This can be thrown into the form dt § * = - • ( y ) g(x) - f(x) * ( y ) (1.2.5) We as sume that x f ( x ) > o f o r x +- o, f(o) = o; g(x) > o, <V (y) > Furthermore,it o and y • (y) > o f o r y $ o,4>(o) (1.2.6) = o i s assumed that the r i g h t hand sides of (1.2.5) s a t i s f y the conditions guaranteeing the existence and uniqueness of solutions of (1.2.5). We construct the following Lyapunov function f o r the system (1.2.5) y V(x,y) = | f ( x ) dx + j - ^ x o y ) dy o Clearly,V(x,y) i s p o s i t i v e d e f i n i t e . Let us compute i t s t o t a l time d e r i v a t i v e i n view of equations (1.2.5). 13 § = f(x) y ^ + " = y } ^ s [ - <P (y) g(x) - f ( x ) * ( y ) ] < ° 3 y * ° f o r ° for y = o It i s easy to see that y = o does not contain a p o s i t i v e h a l f t r a j e c t o r y of the system we now x except the o r i g i n . I f ( 1 . 2 . 5 ) assume that y f f ( x ) dx — » f o r |x| -» «*» * ; o j ; j j j r ^ dy-*-*>for o |V\-* *> ( 1 . 2 . 7 ) 7 then V(x,y) i s i n f i n i t e l y large. Thus we prove the following Theorem 1 . 2 . 6 . I f the conditions ( 1 . 2 . 6 ) and ( 1 . 2 . 7 ) are s a t i s - f i e d then the t r i v i a l solution of Q - . 2 . 4 ) i s asymptotically stable i n the l a r g e . It may be remarked here that the construction of s u i t - able Lyapunov functions i s possible i n a very small number of amples (see [ 3 ] , [ 5 ] , [19], [24], [27], and ex- [ 2 9 ] ) . In 1 9 5 0 Erugin [ 8 ] proved the following theorem f o r the system of two equations ,i.e., f o r § = P(x,y) ( 1 . 2 . 8 ) Theorem (i) (ii) 1 . 2 . 7 . (Erugin) the point We assume that ( 0 , 0 ) i s the only point of equilibrium, the unperturbed motion x = o = y i s asymptotically stable and consequently any motion started i n a c e r t a i n region ( <: ) 14 x 2 + y 2 * * (1.2.9) possesswthe -property x ( t ) —» o, y(t)~-» o as t -» *> , (iii) (1.2.10) a straight l i n e L(o,°») going, to i n f i n i t y from the point (o,o) i s intersected by the motions i n one d i r e c t i o n only f o r t —»«•», (iv) (v) the motions having bounded polar angles are bounded, there are no periodic motions; then a l l the motions possess the property (l.2.10). The above theorem has been generalized to the case of a system of n equations by V.A. P l i s s [26], 2. S t a b i l i t y i n the large of the system ^ = F(x,y), §fe = f ( «* ). In t h i s section we s h a l l consider the system of equations i |f = ^ (2.1) f(*'), ^ = cx - dy where c and d are constants, c * o ; the functions F(x,y), f ( ) are continuous and F(o,o) = o, f ( o ) = o. Besides, the f u l f i l m e n t of conditions of uniqueness of the solution x = o = y i s assumed. The above system was considered i n the works of Ershov ([6], [7]) and Krasovskii [19]. Following Krasovskil we transform the system (2.l) to the following form (2.2) by the change of dependent variables expressed by the r e l a t i o n s «•* = cx - dy 15 The above transformation i s non-singular because c i s not assumed equal to zero. We then have If -'• where * («*,y) = c F( (2.2) r + d y , y) - d t(f) . c Krasovskii constructed the following Lyapunov t i o n f o r the system (2.2) V(«-,y) c f(«*) 7 «* [•(«* J y j * ( o , y ) dy o proved the following theorem.* I f the conditions o y+(o,y)< and •* jf(«-)dro = and using Theorem 1.2.5 Theorem 2.1. . func- * o (2.3) for y * o (2.4) f o r «* * o (2.5) (2.6) for o ,y) - * (o,y) ] f(r)dr = ~ , <• o j+(o,y) dy = oo are s a t i s f i e d then the t r i v i a l s o l u t i o n x = o = y of system (2.1) i s asymptotically stable i n the large. I t may be remarked that conditions (2.3) and (2.4) can be replaced by the following conditions *. * f(<r) * o y *(o,y) > Ershov [7] o f o r «* * o (2.3) for o (2.4) y * claimed that Theorem 2.1 holds without the requirement of conditions (2.6). In fact,he stated the following 16 theorem: Theorem 2.2. I f conditions (2.3), (2.4) and (2.5) are s a t i s f i e d f o r the system ( 2 . l ) , then the t r i v i a l solution of system (2.l) i s asymptotically stable i n the large. The following example shows that conditions (2.6) cannot be removed i n general. Example. Consider the system of equations (2.7) where f ( x ) i s defined as below ,-2x f (\x )/ Obviously, xf(x) > y *(°,y) = T-| = — £ — .% 1 + e"" + 2 — e~x f o ~-2 1 o = for x * r x for x >, < 1 1 o and f ( o ) = o y(-y -f(°)) = - y < o f o r y * o • " [ * ( ,y) - *(o,y)] = x(-y -f(x) + y) = -x f ( x ) C o r for x * o Moreover,it i s not d i f f i c u l t to show that f ( x ) i s continuous and s a t i s f i e s the L i p s c h i t z condition. Thus a l l the conditions of Theorem 2.2 are s a t i s f i e d . We show that the t r i v i a l s o l u t i o n of this system i s asymptotically stable i n the sense of Lyapunov but not i n the large. The s t a b i l i t y i n the small follows from ttee f o l lowing Lyapunov function x V(x,y) = J" f ( x ) dx + i y \ o ? 17 We now show that there e x i s t t r a j e c t o r i e s going to i n f i n i t y f o r t —* + 0 0 . . It i s easy to v e r i f y that y = -e g r a l of the system on the i n t e r v a l 1 £ x < i s a particular inte0 0 passing through —1 —3 the point (1, -e ) at t = o. We show that along the curve y = -e dt > o.i.e. x increases with the increase of time, ' cix dt / \ = " T —2x —x ~x = ' TV^-x +E F(X) = > o for x >,1 We integrate |x = ~ along the t r a j e c t o r y y = - e " and have dt 1+e~ x t x t J + e " dx = J dt or e + x |= t | 1 o 1 o e x x x 1 X e _ x X or x + e x - e - 1 = t Prom the l a s t equation i t follows that as t — • + <x> , x —-» + 0° , i.e.^the p o s i t i v e h a l f t r a j e c t o r y y = - e ~ of the x system (2.7) tends to i n f i n i t y as t —-* + »» . Hence i t follows that the t r i v i a l solution i s not asymptotically stable i n the large. 2.2. Let us consider the system dt •ff dt (2.2.1) = f(r ), = cx -dy, c * o under the conditions: F(o,o) = o, f ( o ) = o > o f o r «* * o «* [•(« ,y) - *(o,y)] <• o f o r «- * o # (2.2.2) (2.2.3) (2.2.4) 18 y * ( o , y ) 4> (c,y) where < o F(JLL&L, = c * for y y) - o (2.2.5) d f ( •• ) (2.2.6) c Besides, the the f u l f i l m e n t of conditions s o l u t i o n x = o = y i s assumed. (2.2.1) to the following As before of uniqueness we r e d u c e the of system system jf - * (2.2.7) We c o n s i d e r the right-hand sides = o a n d f(o*) f(«*) =0 the positions of (2.2.7) = o. Since o n l y when * o n the («*,y) «• f ( « * ) > = o,i.e of the curves o f o r f ( «• ) # J We n o w t u r n t o t h e c u r v e r e p r e s e n t e d 4» ( 0 , 0 ) that <*»(«f,y) < * (o*,y) these o f o r f ° 0 ff* 7 o , y = r y Thus d «• dt o . From these <t» ( * * , y ) > made > o «fr ( « * , y ) = . 4»(*',y) < o (*",y) i t changes > It i s easy to the curve see o f o r y o. In of the curve o, < deriving (2.2.4) and second i n the that lying lying «> ( r - , y ) o, 4> («*,y)<o_, = o lies <P ( » * , y ) f o r the points < sign i n the 4> ( » * , y ) f o r the points . o f o r < o, the y-axis. follows that «* < o , y t h e curve of d «* -rr— * > = o . We o b s e r v e use of the conditions and hence and f o u r t h quadrants. = 4> ( « * , y ) facts o for i t follows that = <p ( « * , y ) 4» ( « * , y ) ; on the y-axis we h a v e and f o u r t h quadrants second o > o and 7 conclusions (2.2.5). «* > o represents by by p l a n e , i . e . , o f <P (a^y) <r* + o a n d f ( o ) = o a n d o n t h e «* - a x i s represented to the left = o to the right = o 19 •^2. f ( «*) = o > f o r the points lying to the r i g h t of y^axis = f (•*)<, o f o r the points lying to the l e f t of y-axis The pose the plane axis for y curve tion «* > 4> ( « * , y ) curve (<f,y) cos 8 = into o for = o a n d r < o y(t) ; *(t) , <s* «* <. o . cos 6 8 = - *» s i n s and The polar direc- co-ordinate then = of r decom- i s maximum o n t h e i n f i g . 1. We i n t r o d u c e sin $ axes i s maximum o n t h e y - o* > o a n d m i n i m u m f o r r signs the co-ordinate s i x regions. indicated , y = r and The («*,y) o and minimum f o r y of motion i s = r * 6 + y + y sin 8 cos i n different regions are given as below: r maybe (1,4) >^ o , > 6 (3,6) The L y a p u n o v o r < f may b e (2,5) o, d may b e V = Obviously, Now iLfl dt total f(«0 > - o ^ o (2.2.7)is r r - J * ( o , y ) dy o derivative ^•(••,y) - * ( o , y ) V ^ y ) is = <*(o,y) time Q y = J f ( c ) d o Its o, . f u n c t i o n f o r the system f V(c,y) \ positive i n view of ]<.o for * o = for •* * o o definite f o r «* = o a n d i t equations i s and V i s different (2.2.7) of negative from zero i s sign. unless y = 20 This means t h a t trajectory the the (ii) of ( i i i ) there the appearing are is the sense Thus is the of we h a v e only point motion are conditions of no periodic a Lyapunov «• p o s i t i v e i n only, Theorem satisfied bounded p o l a r dition the than any o t h e r of origin. positive Whence Lyapunov of half follows the t r i v i a l shown equilibrium, asymptotically stable according Lyapunov, direction 1.2.7 in (2.2.7). ( 0 , 0 ) contain (2.2.7) stability unperturbed structed Since o does not system system point the to one the asymptotic solution (i) of <?* = are angles to Theorem be since (2.2.7) for is con- function. half it motions, axis can be 1.2.7. except are is intersected taken Thus the a l l in the the FIG.i straight the to realize motions line of motions We n o w i n d i c a t e order 1.2.7. the conditions fourth^i.e., bounded. imposed for by what the i n L(o,oo) Theorem having additional fourth con- 21 Let M(t) to started the This in follows the tends origin Q A'. the leaves these > of the This the bounded p o l a r condition Condition B. with o < thus * (o%y) called this either but region to cannot impose satisfies starting in the increase of that region the After the (l). en- y-axis go the tends to the following such conditions regions (l) called and (4) time. there regions the crosses motion either region. i t Any (6). and e n t e r s or i n the this i n led guarantees angles We a s s u m e f y-axis, We a r e condition region motion M(t) entering regions a region,, say entering this We a s s u m e t h a t motion in or that along o. start (6) fact (l) infinity A that out from the since another region goes region to Condition with the origin or tering or a motion M(t) (l) are and no (4). motions We impose B. that * (•*, c) = o has a solution for a l l c. If the region (2) with if * the curve can neither it can Condition A is We n o w s h o w (2). the («',c) increase = • o has the the motion must ing can be carried shows that time i f solution the line intersect that the = t > curve in the regions there are no o • is leaves entering < o in i n region (<*%y) = (3), motions (4) the this (2). 0. and region In fact intersects region (2) region nor Therefore Similar (5). enters the satisfied. t ^ c,since 6 motion M(t) c ±.e , ±f y = c motion after y the motion M(t) a l l the out then Condition B for o r i g i n because the analysis a of ( « * , y ) = 0, cross enter satisfied The with, bounded p o l a r reasonabove angles 22 in the regions bounded p o l a r as angle is The the polar occur are bounded. (2) and (5) This can also > > 6 o. if 2.2.1. either then the the in Thus that the the motions regions (3) any motion w i t h If t r i v i a l imposed with there be of this if with we a s s u m e c o n d i t i o n as results or a view to no m o t i o n s in (2.2.2) Conditions solution are achieved these C o n d i t i o n s A and B with and and (6) bounded the ensure bounded that is in these Condition C. following: (2.2.5) are satisfied C o n d i t i o n s A and C are (2.2.l) that asymptotically and satisfied, stable in large. It may be remarked that dent,i.e., if Condition B holds hold versa. or sides vice of Example 1. In the (2.2.1) system Let us consider Consider the (2.2.5). us for the sake of t h e n C o n d i t i o n C may o r c a s e , i . e . , when linear, however, a few examples = the both indepen- may right not hand hold. now. - y J + -x e - -1 1 (2.2.8) = this We v e r i f y t h a t and C are system dt Obviously, Conditions B linear are dx — dt Let only We c a l l We n o w c o l l e c t Theorem and (5) can C o n d i t i o n B was angles. 8 they and (4) bounded. regions regions (2), angles p r o v e d above polar in (l), system i n this x X satisfies case definiteness the conditions Condition A is assume that the (2.2.2)— satisfied. motion M(t) 23 is i n the fied. Then the infinity y(t) of region as finitely large first quadrant true. verify that It is the re and in + = -x = 6 the b i l i t y Example - i n (2) and i n the large of 2. by writing can be x for , the + y = for + e -1 the This = dt " 0 not c intersect (see fig. the curve 2) However C o n d i t i o n C h o l d s . = - s i n a (-y+e -1 ) + x x 2 8 establishes trivial > € > the s o l u t i o n of In cos 9 0 asymptotic sta- (2.2.8). equation • Cf£) g(y) + thrown i n t o § a l l ) + cos x 3' FIG, 2 c does cos 8 (5). + y _._ satisfied. Consider the equation i n - assumption = o (-y 9 regions f£ This not sine 3 in line 1 x satis- this d i f f i c u l t to e - is a not which our straight y Condition A is equation remain not the to -^becomes and hence Condition B fact f i r s t cannot Hence goes but negative M(t) that and d u r i n g follows that that not •* From the means is We a s s u m e motion M(t) t —* —» — . (2.2.8) (l). - *U) the g(y) f(y) (2.2.9) = o system - f(y) dy_ dt We a s s u m e that the following conditions are fulfilled: 24 y t(y) > o, f ( o ) = o and either h^(x) > o for x * o and g(y) > o for a l l y or h^ (x) < o for x * o and g(y) < o for a l l y where h.(x) = *( ) x for x * o x 1 The Lyapunov function f o r t h i s system i s V(x,y) X y J xdx + J f ( y ) d y = o Then V = o x ( - * ( x ) g(y) - f ( y ) ) + f ( y ) x = - x * ( x ) g(y) I t i s not d i f f i c u l t to see that i n this case Condition C of Theorem 2.2.1 i s s a t i s f i e d . Thus i f Condition A i s also sat- i s f i e d then the t r i v i a l solution of (2.2.9) i s asymptotically stable i n the large. Example 3. Consider the system = 4>(x) £ ( y ) |f = dt •^or the system f (x) 1 w (2.2.10) the V function i s x V(x,y) y = J f ^ ) dx - J f ( y ) dy 2 o Tnen We subject V(x,y) (2.2.10) + = o f (x)*(x) 1 (2.2.10) to the following conditions: <•> (o) = o , h3(x) < o f o r x £ o , where h,(x) = ti$)for x£ o x J (2.2.11) f 1 (o) = o,xfj (x) > o f o r x * o , f (o) = o and y f ( y ) < o f o r y * o 2 2 25 The t r i v i a l solution of (2.2.10) i s thus asymptotically stable i n the large i f i n addition to (2.2.1l) the Conditions A and B or A and C are s a t i s f i e d . Example 4. Consider the equation of the second order ^ (2.2.12) . g < y ) - o + We write i t i n the formof jbsystem of equations |f = - f ( x , y ) x - g(y) dt X We assume that g(o) = o, y g(y) > o f o r y * o and f( »y) > ° f ° x + o x r It i s easy to v e r i f y that asymptotic s t a b i l i t y i n the large holds i f Condition A i s s a t i s f i e d . 3. The Problem of Aizerman. In 1949 Aizerman [ l ^ proposed the following problem. Let there be given a system of l i n e a r d i f f e r e n t i a l equations dx y 1 dt" = h dxi 1 a i J J x + > dt~ = t fr — i a i j X J (3-D ( i = 2,3,-.- n) f Suppose that f o r the given constants a^^ ( i = 1 , •;«..«-n,« j=1 ,».»jn) > and f o r an a r b i t r a r y value of 'a' from the i n t e r v a l J. < a <. P a l l the roots of the c h a r a c t e r i s t i c equation of (3.l) have negative 26 real parts. Let ax^. b e dx x^ = Xg in the = is • large (3«l). We t h e n have n j ~= = " 1 ^1 j J r X + ( k) f x (3.2) " ± "dt It i n 1 dt" dx r e p l a c e d b y fCx^) j = = 1 «U X j ( required to find = x system = n or not, o of out = i whether is (3.2) for«arbitrary 2 choice " - - n the ) t r i v i a l solution asymptotically of^continuous stable function A f(x^.), which reduces to zero for x^ = o and which satisfies the inequality L The answer t o The interest above is the above i n the of a.. Hurwitz what problem is values s o l u t i o n becomes here the to i n the the i n above * we questions. negative. If the under made o n 2. The the [28]. i n the First of 3, trivial the s o a l l we t a k e what affirmasolution generalized = o), that large. k = the Por i n the and f ( o ) ^(Xjj.) case n = f25j). (Pliss s l i g h t l y change problem i s large (3.3) o negative c o n d i t i o n (3.3) s h o u l d be when n = solved by P l i s s we c o n s i d e r the ^ following asymptotically stable case for 2 i n the the ( i . e . , the assumptions x . revived i f ourselves answer conditions additional pletely k asymptotically stable 1 < P f (x ) problem i s the nt) and f o r not cuss ^ problem and ask values tive < xl the n t r i v i a l We s h a l l 2 has = what been 2 = k, discomi.e., system f = ax + , f ( y ) (3.4) dt = b x + °y 27 under the assumptions that f(o) = o , a + c < o , a c - bh(y) > o^y * o (3.5) where h(y) = f (y) for y * o y Besides, the uniqueness of the t r i v i a l s o l u t i o n i s assumed. System (3.4) was considered by N.P. Erugin [9} and I.G. Malkin[2l] . Malkin [2l] showed that the t r i v i a l solution of system (3.4) i s asymptotically stable i n the large i f f o r s u f f i c i e n t l y large values of Jy| the inequality ac - b h(y) > 6 holds. This condition can be relaxed to the condition that y J (ac - bh(y) y dy — » +°° as | y\ —» + «* (3.6) o We show that asymptotic s t a b i l i t y i n the large of the t r i v i a l s o l u t i o n of (3.4) holds without the requirement d i t i o n (3.6). Por b = o, from (3.5) follows that a < o, of conc< o and an immediate integration of the system (3.4) shows that the equilibrium i s asymptotically stable f o r a r b i t r a r y i n i t i a l d i s t u r bances and f o r ^ a r b i t r a r y choice of the function f ( y ) . Let b * o. We introduce new variables defined by then | i = "it b = dx . at a = x = y = c| = b x b ( a x + + bx - ay y f ( y ) ) - a(bx cy) = -y(ac-bh(y)) + °y = ' x + a y + °y = ' x + ( a + °) y' 28 Thus (3.4) t h e system dx dt reduces to the following = -y(ac - b h(y)) = x + We r e p r e s e n t (a + c) the curves (3.7) y -y(ac - bh(y)) obtained by putting the right hand sides the these represents the straight y (x,y) plane. The f i r s t = o and the second see of the straight of = o and x+(a+c) line y = (3.7) e q u a l a + c . It t o zero on line i s easy that dx dt dx dt dt = -y(ac - bh(y)) > o below the x-axis = -y(ac - bh(y)) < o above x-axis = x + ( a + c ) y > o below the straight the line x + ( a + c ) y = o dy. dt = = x + l a + c) y above Co x The d i r e c t i o n of motion i s x + ( a + c ) y = o axes divide regions. x = r r <t> , y = x cos * = - x s i n 4- ficult the the plane sin + f cos r of f may s i x co-ordinates = r sin to v e r i f y that signs (1,4) co-ordinate (x,y) into + y . and . It i s not i n different * > o dif- regions a n d tp a r e g i v e n a s b e l o w be ^ o, (a + c) line y = o i n d i c a t e d i n f i g . 3. T h e s t r a i g h t We i n t r o d u c e p o l a r cos Then r and the + the straight ; y=o line to 29 (2,5) r < 4> m a y b e o, The axis of x. This the region there increase cross of time is (3.7) may b e i n the region = x + ( a o, must (6) + c ) y > / d > After entering x = I seen > cross o. the = x + o (l) i t cannot ( a + c ) y = o the straight The L y a p u n o v remain with the line function f o r the sys- to be 2 2V line intersects f o r a l l I. easily the region the straight since x + ( a + c ) y = o tem r follows from the inequality (6). and must (3,6) o motion started Jf in \ r y x + 2j (ac-bh(y)) y dy o Repeating arrive at the in The t r i v i a l i n the large It the This Since The i t is as i n Theorem solution of under Conditions may b e n o t e d the regions we d i s c u s s argument 2.2.1, we following T h e o r e m 3.1» stable t h e same (1) i s asymptotically (3.4) (3»5)» we m a y o r m a y n o t h a v e a n d (4). B e f o r e s t u d y i n g the case * > £ > o n = 2, k = 1 system system quite conditions that system i - f - -«W r - » « (3.8) is a p a r t i c u l a r case an important to which x P(x) > (3.8) o , system, we d i s c u s s i s subjected x g(x) > of the system o ±t (2.2.1). independently. are f o r x * o (3.9) P(o) =o = g(o) 30 The V - f u n c t i o n i n this case is x 2V = y H and = (3.10) o y(-g(x))+ g(x)(y - P ( x ) ) = - g ( x ) The trajectory + 2 Jg(x) d x 2 straight of the line system x = o does except (3.8) not P(x) < o for x * o = o for = o x containo-positive the origin. In fact half J^x = y dt for x = cally o. Hence-it stable i n the for the the origin is the see i n virtue of The system, = y - — = y of there exists of equilibrium. It Conditions P(x) to the left of c o to the right -g(x) ^ o to the left = -g(x) < o to the right The r co-ordinate d i r e c t i o n of co-ordinates = x and cos -x r may be ^ o, 4> < (3,6) r may b e ^ o, 4> f> = r + y regions o < (2,5) o. is the of of we n o t e not axes is curve the the of motion sin + + y i n the (1,4) x Also V that difficult to that (3-9) > o and the asymptoti- a Lyapunov f u n c t i o n cycle. = polar solution is only point r * = signs Since trivial any l i m i t r The the be _ p(x) regions. We i n t r o d u c e that cannot the |f six small. there curve y = P(x) into follows = curve y P(x) = P(x) y-axis the y-axis divide the plane represented c o s <p , y = r sin y (x,y) i n sin 4 fig.4. . Then • c o s * are r g i v e n as < o, <fr below : maybe >^ L o 31 Let us f o l l o w the M(t) a f t e r it tive half that y = A intersects y intersects y-axis. motion the Let us nega- assume the curve = F(x) for a l l A,i.e.,F(x) has a s o l u t i o n f o r a l l A, = A then a since i n the r e g i o n ( 4 ) 4» and y is the motion M(t) cannot oLM. LM increasing remain It definitely cannot i t the region (5). the region ( 6 ) . This Since m o t i o n must argument there occur must cross leave (5) i n the this + y >/ region 'Thus bounded. tive half satisfied Theorem a i t y-axis. 3 . 2 . solution If the for d > (l), with ( 2 ) a n d (5), straight Thus a n d we h a v e either fact then and enter the origin r < o i n ( l ) . ( 2 ) and ( 3 ) .This angles L ( o , 00) conditions Similar shows that i n the regions angles can they taken Theorem the ( l ) , only are bounded p o l a r can be of this angles posi- 1 . 2 . 7 are theorem! ( 3 » 9 ) are the with or ( 6 ) , the and as proved above the following a l l A, that bounded p o l a r line Conditions enters the region a l l motions a l l the y = F(x) o i n the region with bounded p o l a r follows that Fox' t h e i t curve and enter f o r the regions regions the follows from the -F(x) a r e no m o t i o n s bounded. has = F I G . 4 cross In the region dt holds region ( 4 ) and ( 6 ) . The motions (3), are o i n the bounded and o L . Therefore region. < trivial satisfied solution and F(x) = A of (3.8) i s 32 asymptotically stable We r e m a r k the large holds i n the that large. by using T h e o r e m 1.2.5 the stability i n i f x J"g(x) d x —* » a s | x | —* oo o This is required i n finitely . order to discussed 2 at* be the V function in (3.10) i n - large. The w i d e l y c a n be make dealt i n transformed the + equation ( ) dx + g( ) o dt f x x same w a y a s the system (3.11) = (3.8), (3.1l) since can to X = Jf g(x) where P ( x ) = j" f ( x ) dx o We s h a l l now d i s c u s s the T n = 2, k = 1 ,i.e.,the Aizerman p r o b l e m f o r the case system If = f U) + ay (3.12) f t under the = b x + C y condit ions : f (o) = o ; c + h(x) < o j c h(x)-ab >o (3.13) for x + o where h(x) = f(x). x * o x This and Malkin system [2]^ . It is discussed i n the was p r o v e d b y E r u g i n works that of i f Erugin c~ + ab ( t- t 8 j , [13] ) o,then 33 the trivial large is under quite become does not Erugin 2 + so clear hold also ab (3.12) of (3.13). conditions lengthy, w i l l c solution from our under showed = o, it lim that proof for - proof here a why the the stable by Erugin short (3.13) proof of of i n the case i n c i n the in lim —> the + abx) f(x) dx - + c f(x) - 1= abx abx) dx - c f(x) + (17) as well the case by P l i s s as sufficient. where the showed t h a t The question stability o. i n the case + (3.14) J in the (3.14) conditions of the large region does L + «° abx .o Krasovskii = that -j (c It large ab large x -< * > j theorem theorem. r x the this the stability stability sufficient f(x) asymptotically The conditions is J(c x-»<* we g i v e is of are necessary stability not hold = o, then is in discussed [25]. 2 We a s s u m e integration of the that solution x troduce new dependent If- |£ a = of * o. the Let a stability (3.12). system in Let the immediate large now a * o. of = a - abx x = x y = ay f(x') = % dt - c c + y' + dx dt f(x) = - cx c x' a(b x + cx) - c (f(x) + the We i n - variables dx dt t ab yields o = y = d t + system t r i v i a l Then c ay) 34 The system (3.12) i s reduced to the form H = f ( x ) + cx + y § = - x(c h(x) - ab) Comparing, i t with system (3-8) ® have w F(x) = - x(c + h(x)) and g(x) = x( c h(x) - ab) The curve y + cx + f ( x ) = o and the co-ordinate axes divide the (x,y) plane into s i x regions. The d i r e c t i o n of motion i s repre- sented i n f i g . 5- We consider the following cases. Case 1 c < o. Consider the straight l i n e 2 , y = - ° + " a If c + x (3.15) ab > o, then the straight l i n e (3-13) l i e s i n the f i r s t and t h i r d quadrants. Let us see how i t i s situated on the (x,y) plane with respect to the curve y + cx + f ( x ) = o (3.16) r IG. 5 Let y ( ) and y ^ ) denote the ordinates of (3.16) and (3.15) 1 2 respectively, then (1 y ) (2) - y 2 t ab x c 2 2 = -c x - c f ( x ) +{c + ab) x = -x(c h(x) - ab) ^ c c = - cx - f ( x ) + £ according as x ^ o, 0 35 i . e . , the first (3.16) curve quadrant lines y that A and the the = t r i v i a l large above i n the ab < the third 2 = - y + c c = - a is to quadrant.. f(x) intersect asymptotically o i n the regions (l) and no prove and line y for the follows a l l stable A. i n c + a Hence the 2 = - the x p c such conclusion that in this as case which ensures (4) i n whence 3.2. straight However,we can (3.15) Obviously, intersect, Theorem the line x P cx - (3.12) o,then straight and f o u r t h quadrants drawn. < - 6 curve second can be <j> < ity c + i n the the case according 2 If lies = A and y s o l u t i o n of i n this above and b e l o w i n straight y lies the stabil- large. According c + to h(x) i.e. (3.13) Conditions < o, h(x) < c h(x) -c, - h(x) ab > o for x * o are satisfied < c 2 but since take c h(x) < + ab < -c- f(x) o, the f(x) = - can cx above inequalities t h e n be - J.(x) written where t(x) us calculate r 4> = * the = - - S i n * (y regions + cx s i n 4 > ( y - X»(x)) - (l) Jfc. o according as : sin<*»(f(x) = = in <<> as o X Let i f s i n < ^ ( y - X»(x)) - X.( )) + and X (4). + y) - - cos^> ( c h ( x ) cos*£c(-cx + COS* c o s 4> £ ( c ^ + [ 2 ( c +a b ) x + - - ab) - £(x)) ab) x + x abx] cL(x)j ^. € < o we 36 2 Case 11 c > o. In t h i s case c + ab < o,because from Conditions (3.13) < h(x) C - c c as h(x) = - c - JC(x) Let h(x) be taken ^ ^ c < - c -i(x) < c + ab _x(x)< c o < - c + ab > c 2 or x. ( ) - ° Let us consider the region ( l ) . I f that - >o x o < JC (x) <. - c 2 + a b c <t> £ - £ < o, then i t means -~[(c + ab) x + c x X ( x ) l = o, i.e. l i m 2 X—*<*> l L J i xjC(x) = <*, m x- whence i t follows that the straight l i n e y = A i n t e r s e c t s the curve y = - cx - f ( x ) f o r a l l A In f a c t , - A = cx + f ( x ) = cx + (-cx - xX(x)) = - x-X(x) therefore x £(x) = A I f x £(x) = A does not have a s o l u t i o n f o r a l l A, then ™ x— x<(x) = 1 lim X— J( 2 c + a b ) + x finite = D cxX(x)l^li!H f-(c + 2 = l i E [-x(c X-»«o~ whence follows that Case 111 c = o. <p ab) x ] 2 + ab)] - <, -fi< o. The System (3.12) reduces to + l i E (_ c X ( C X J xjl(x)) C(x)) , 37 £ and the tion this , a l l these 3.3. Theorem solution is If to h(x) <o, 2 2 = which is = - V function satisfies bining : the — This reduce s y s t e m we t a k e 2V d e r i v a t i v e of ••»« (3.13) Conditions Por the • abx , ab a l l x + following o. Lyapunov' • func- y 2. / \ h(x) the results we h a v e for System the ab < conditions the i n the ab Com- theorem: * large 1.2.5. Theorem following (3.12) c2+ asymptotically stable of o, then the t r i v i a l under Conditions (3.13). the line proof y neither y + (4). cx = In the of stability say ° 2 + c that + f(x) = case a D x the o for c 2 i n + ab = o, we r e m a r k the Case 1, coincides straight with line a l l A nor we f i n d the y that c that x-axis o the and hence = A intersects 4, ^ . ^ ^ o i n < the the regions and f r o m straight we can curve (l) and 38 4. A generalization we d i s c u s s the of system the problem of Aizerman. of d i f f e r e n t i a l dx dt ax equations In this section given by + ^ ( y ) (4.1) f (x) dt under the where h. (y) t r i v i a l cy conditions: and h „ ( x ) are The obtained + 2 above certain system theorems solution x defined by f. (y) was f i r s t regarding discussed We p r o v e and f ( x ) = the 9 [l&jwho i n the large here xh (x). 9 by Krasovskii the s t a b i l i t y (4.1). = o = y of = y h, (y) of the following theorem: T h e o r e m 4.1. or h^(y) < tion of Proof. If either o for y * (4»l) is (4.2) follows the h. (y) > f i r s t that o, and t h i r d y * o, < o, quadrants. + cy = o. We f u r t h e r above the curve f (x) + note c < the then 2 the t r i v i a l i n the large under x * o. We c o n s i d e r (x) + cy = o curve T h e same that o, and hg(x) > o, + f.j ( y ) = o a n d f f^(x) p stable a < o, o, y + o and a o for x 2 h^(x) < o f o r x * h^(y) > ax Since o f o r y $ o and h ( x ) > asymptotically We a s s u m e ditions o, h^(y) > i s the curve cy = o i n the f i r s t the f o r the Conditions Prom Con- curves = o lies i n curve ax + f ^ ( y ) quadrant solu- (4.3) ax + f . ( y ) true o. $ o = o and below lies i n 39 the third the curves - " 2 y *1 quadrant. in In fact, i f y^ and y respectively, (4.3) h (x) -ax ^ (y ) 2 is ac-toi(y) !— -x f 1 easy to see the right cy < o to the left + cy > o to the right f (x) + |£ = f (x) + f ^ y ) = x 2 function x(t) f (x) of are r < (3.7) r (4.8) r < cos o, of according as >• o < o; o for y and the > o motion are x of of of = r the curve the the ax curve curve the cy functions ax + fr,(x) 2 + f^(y) y is < o. = y = r sin $ . o o = o = o o maximum The shown i n f i g . 6 . A s b e f o r e c o s <p , = cy+f (x) y(t) and minimum f o r = ax+f^(y) curve curve + f^(y) we Then <p r maybe maybe maybe h?(x) — ^ o x maximum o n t h e a n d 4> i n d i f f e r e n t (l,5) o, ordinates s i n * + y f y co-ordinates + y sin & cy = is d i r e c t i o n of polar signs + 2 and the cos«fr gions 1 2 curve -x f (y) o and minimum f o r introduce (2,6) to = curves The o § the r*= < ax on r left = > of the ft y (y) to ax for N \ >o = The the that § + denote then c It 2 > y<i 4> o, ^.o, <. re- / o >^o <* > maybe Prom Theorem 2.1 o o of Erugin's w o r k [8] it f o l l o w s that there 40 is at least of e q u i l i b r i u m , i n each started i n one the the regions and (8)). L e t integral regions (3), us of or (l), suppose the motion leaves Let the least ac - (y) + CgX cy above in be 2 o will o. is easy It curve f i r s t versed. ax = the the > f (x) tersect quadrant. = the + 2 We c o n s i d e r + f^(y) o the this cy In the and the straight to o the to the r < o, 4> m a y b e h (x). c 2 that this and below the third quadrant positive y = half t,since (4) (l). We of time. otherwise straight straight line curve + f^(y) = ax It = o„ are re- motion cannot i n - the The line lies positions L, < o. $2. the enter in >^o region finite the y-axis. or increase We c o n s i d e r r e g i o n bounded by y line is 2 point motions origin the the only Other go region with hold. the (8). motion enters verify = origin, and (4) (since (8) not the either upper bound of h (x) 2 regions (8) that show that c going to the (4) or (5) curve curve cannot go to the dt origin,since ax + f^(y) gin with line x = f (x) + 2 origin (7) = the o o. Therefore and enter increase the of i t region time. Por t a n d s i m i l a r l y show t h a t cy as which = o t—» and enters . + oo completes In the <t> <. the the must proof the to we take the straight motion crosses the curve (2) prove curve goes (3) where hold Theorem same w a y we c a n the i t region region of where (2) Similar reasonings the intersect the for it goes the the to o r i - the regions (5) 4.1. following- theorem system (4.4) TjJ = bx + cy for and 41 voider the conditions: h.j(x) + c < o , x * o, c h ^ x ) - h ( y ) b > o, x * o, y * o, 2 where h (x) = 1^1 , x * o, h ( y ) = Mil x Theorem 4.2. 2 , y * o (o)=f (o)=o 2 (4.5) y I f b h ( y ) > o f o r y * o and the Conditions (4.5) are g s a t i s f i e d , then the t r i v i a l solution of system (4.4) i s asymptotic a l l y stable i n the large. > 42 5. The stability using of the we c o n s i d e r under which we w i l l be 2. s y s t e m Jf = F(x,y), the system working will be B . A . Ershov(6] d i s c u s s e d this system theorems regarding We h e r e = f Jf (•") methods. section solution. are large this Section tain the qualitative In tions i n the stability show t h a t his in results different the are (2.l). The assump- from those and o b t a i n e d large of correct the but of cer- trivial the proofs wrong. Let us consider the system (5.1) = Jf and c^ , d^ are f(«0 positive continuous function, respect x to and y , where «- = c.,x constants. for * discuss of F(o,o) three x, 41 > assume y. that cases F(x,y) derivatives Further is a with we a s s u m e = o that (5.2) ( i ) | ^ v (iii) d.,y order p a r t i a l a l l values o, - We s h a l l having f i r s t 1| We s h a l l , / 3x < ° j ( i i ) 5 Z -&X = o, o. ^x The side of the continuous second function f(«"), equation of (5«l) is appearing in the right subjected to the following conditions c f(«*) f(o) > o = >/ o 9 6* for «- $ o (5.3) (5.4) o for a l l (5.5) hand 43 In the condition i s not (5.5) 2i£ taken to be identically aszero . The equations § -(-if) (5.1) c a n b e x + (-rf) written y as ^ + (5.6) dt 'x=o,y=o Q J \°J x = 0 ) y = 0 where \ ° /x=o,y=o ^ „ , (x \ »«V X x=o,y=o x = F(z,y) -(^f ) Mx.y) - (^|) y ' x = o , y = o V y ' x = c , y = . a, ( O f )y . ,(0-.^) « y) ' x=o=y We w r i t e f o r s i m p l i c i t y = a, \ * 'x=o=y The system three =b x 0 c a n then be w r i t t e n i n t h e dx dt = |f = d / ;x=o=y form - N a x - b y + °y(x,y) cx thus (5.7) - dy + <y ( x , y ) correspond to the values 1, o, -1 o f N in respectively. (5.7) Case x=o,y=o (5*6) cases ' x = o = y = '1 The y 1 < _2LZ a o. Prom equations |f = (5.7) we h a v e for N = 1 x |f - ax - by + %(x,y) (5.8) = cx - dy + «H ( x , y ) 44 The equations of the f i r s t dx — = approximation are - , by ax - (5.9) Jf The characteristic = cx - dy equation of (5«9) + A a - b c Ti+ = o d 2 or ^ + ( a + d)h + ad+ b e= o The roots of this e q u a t i o n have a + d > o Since the the roots t r i v i a l cording seen (5.10) of and have 1L + + condition (5*5) parts,since be > real o parts, i t follows In JLL d i x 4 i s (x,y) plane. sents the stants, curve straight f = o i s P(x,y) situated By virtue i n the second & e dx These = P(x,y) of Conditions «* = o . situated = o passes ' '> ± line 0 that JLE. < O, d, > Since c^ through the and fourth -Tjyfc considerations and t h i r d origin s a n d (5.4) f ( ** ) = o a n d d^ a r e p o s i t i v e quadrants i o. 1 = o and f(•* ) = o on the (5.5) i n the f i r s t *^\x easily 1 and the f a c t the curves ac- fact, 3x We r e p r e s e n t that asymptotically stable of p e r i o d i c solutions _lZ _ ny (5.10) ' real c r i t e r i o n of Bendixson. jx of ad negative t o L y a p u n o v . The a b s e n c e by using the view negative solution x = o = y o f ( 5 . 8 ) i s M. in i s since since quadrants. P(o,o) the = o. slope repreconThe It is of P ( x , y negative, show t h a t (o,o) i s the only point of 45 (5.8). equilibrium of only point sense of of by Lyapunov,(iii) the angles tions of = |f there motions polar shown equilibrium,(ii) now show t h a t ted We h a v e are motions P(x,y) there exists a is far e x i s t . no straight line only bounded. With this thing for the P(x,y) It is points, = (i) periodic direction fig.7). that asymptotically i n one (see >o it so the origin stable in solutions. motions see shall intersec- with i n v i e w we e x a m i n e easy to the the We L ( o , <* ) w h i c h i s and a l l is bounded the direc- that lying to the left lying to the right of the curve o dx — = P(x,y) <o for the P(x,y) points, = of the curve o |f = f(o* ) >o for the points, lying below the straight line |f = f ( c ) < o for the points, lying above the straight line The f u n c t i o n y < o x(t) on the attains straight y(t) line maximum f o r minimum f o r x < o on the F(x,y) The straight the = curve dinate eight o. P(x,y) axes We f(«") = > o and x maximum f o r y > line (x,y) f(c) o; = o, co-or plane f o and minimum curve = o and the d i v i d e the regions. attains = into introduce FIG.7 = for o o 46 polar co-ordinates Then x = r cos r = x cos = -x r the signs (l,5) r (3,7) r of and ^ o, may be ^ o, a Ershov (3), angle or us i n these either this these could that regions has regions. is cannot > 5 > <p as increases, y-axis main To enter in this (4,8) r of f(«r ) = we <$> are < o, qi > as <j> o, be - <P o or follows : may b e ^ o may b e >^ o. o i n the regions the > 4> regions after (l), o is not regions. >6 q (t- t and hence these i t intersectTo sufficient to bounded p o l a r true In fact, i f ), whence there angle i f we > <t «i , follows w i l l be an i n - regions. intersecting (7). We s h o w t h a t regions = o. assertion remains <*> after i n these curve F(x,y) any motion w i t h r e g i o n and c o n s e q u e n t l y show t h i s < these motion leaves region <j> any motion w i t h bounded p o l a r o i n these a motion M(t) the o increases time when the Let r However,the above that of cos and regions doubtful,since i n t e g r a t i n g we h a v e stant y (2,6) out line then t <p + be <p .. sin and any m o t i o n f a l l i n g to get there show t h a t r s i n <P o cannot straight reasoning guarantee in the <p > there s t a r t i n g there ing + y sin > = [6} a r g u e d t h a t , s i n c e a n d (7) (5) y i n different <p he m y r <$ , the negative motion M(t) w i l l enter half cannot the re- region (8). write = J_ r = —( f-x sin«f> 1 + y c o s * ] (5.11) 1 -P(x,y) s i n <> + f(cy) cos«>)>o i n region (7) 47 If M ( t ) does infinite since ficiently is $ > o it with _]!_£. ( 7 ), the region (8) it the that i n this remain in the x this, x-axis consider = A and the motion cannot since 4> gion (l) r o the <£ and the region A l l this Por the Thus the a l l the motion shows 5.1. trivial the (l) Since of the line the i t region either goes x origin must (2). to as the It J point of for L(o,°o) conditions of we can Theorem take 1.2.7 or o. from The region go go out region motion rest polar of angle positive satisfied To (2). line to i n - <r* = o, the the of origin^ the re- (2),since equilibrium or the the are °© \ leave entering decreases, cannot > than follows the x it therefore straight (l) Since After This enter suf- greater may be tp any m o t i o n w i t h bounded line and For motion must necessarily holds "becomes o. t —»+ Here i.e. in the reasoning the (l). region = A. > assumption. o and the £ taken time and must in and x(t) <r i n c r e a s e s . x, region < then always be our the f therefore that straight to of r e g i o n bounded by Similar (3). following Theorem region intersect enter the region the and goes o, non d e c r e a s i n g increase and e n t e r x-axis. o > is «j> c a n the either the see increase f(v) = increasing which contradicts tersects fact F(x,y) are that and hence w i t h region curve the >, o , follows the cannot the and y ( t ) x, that (5.1l) cross x(t) large assumed from not enters the is regions. bounded. half y-axis. a n d we have theorem: Let J|Z solution 0x of <_ o . (5.l) Then is under conditions asymptotically (5.2) stable - i n (5.5) the large. 48 Case II U! ti x o. = The equations i n (5.7) this case take & - - * • -sty) Jf The equations of first = cx - form (5.12) d y + ~i> ( x , y ) approximation dx dt the are , " = b y (5.13) = Jf The characteristic equation h - c The d real > o parts only = b = o is or of the roots of this o, whence follows solution of system (5.12) F(y) o represents curve o represents common given f, + by d ? i + b c equation > The f ( «•*) (5.13) of = o \ + d and be t r i v i a l c x - d y point of = the straight f(c ) = o. the i n are asymptotic the line sense the w and F(y) = of negative,since stability of the Lyapunov. straight = o. The o. It is line y = origin easy to o is and the see that cLx -rr Q.T/ = ^(y) > o for the points, lying below the x-axis — CLX = P(y) < o for the points, lying above the x-axis dt = i f(c) > o for the points, below the straight line f(°*)= o •^7 = f (c) < o for the points, above the straight line f (**)= o CI t y < o for The function the points y(t) o n f (<**) is = maximum f o r o and x ( t ) y is > o and minimum maximum f o r x for > -o- a n d 49 minimum f o r f («•*) = o divide the The < and o for the the axes of plane into six direction sented in x of i n f i g . 8. different points on x-axis. The curves F(y) = o, an i n - co-ordinates regions. motion i s repre- The of signs regions the are r a n d 4> g i v e n as be- low : (1.4) r may b e (2.5) r < (3.6) r may b e see that stant t shown that enters the the the with Thus the the a l l to the Similar it is straight the following = i t is whence easy it intersects the angle line of is the -i^> »i_ -j^. L(o,oo) conditions theorem; -j dj the time. increase (0,0) is To criterion of / (not we c a n Theorem take 1.2.7 line In of time are f(«0 or be and either (2) enters small,(iii) the (3),(4) regions show t h a t can = o only point of equi- any there mo- exists BendJxson identically the it I region the the i n the bounded. exists As i n Case straight of stable ^ o there applied for (i) asymptotically that x-axis. the c a n be shown that o - the increase o r i g i n with we u s e follows intersects arguments motions, F > f —— + —— dX } y For (6) with (2) bounded p o l a r periodic o > o, motion region librium,(ii) tion region 0 >^ d> > T h u s we h a v e (5). o motion M(t) motion goes and > may b e >, £ dt (3). <*> "^o, when the the region no <p o, In to ^0, equal.to positive satisfied half zero) N x-axis. a n d we have 50 Theorem 5.2. Let ^ ox t r i v i a l solution of > o. = o. Then u n d e r system (5.l) is conditions (5.2) asymptotically - the (5.5) stable in the large. Case III !LE dx The equations ax = - by i n this (5.7) + case take the form -\(x,y) dt J&L dt The characteristic = cx - equation of ^ - - a a c or of roots ing conditions ; If t r i v i a l A+ (5.15) The the will = o We n o w r e p r e s e n t f(o') o on the the + (x,y) f i r s t b have = approximation (x,y) is o d negative real o (5.15) parts under the follow- d - a > o (5.16) be - d a > o (5.17) = y small. = <y (d - a)*+b e - a d = (5.16) conditions solution x dy + (5.14) (5.14) of the curves plane. Since (5.17) and is satisfied asymptotically F(x,y) = o and the > o, _LZ are the curve then stable in straight F(x,y) = the the line o is 3 x situated i n the first given f(«*) o. of by Since e q u i l i b r i u m we w i l l Condition and = the 1. The straight t h i r d quadrants we a r e have curve line Obviously and to interested impose F(x,y) f(ff*) = condition and = o so the i n having an e x t r a is is situated straight a unique line point condition: between the x-axis o. 1 implies the condition (5.17). It is 51 not d i f f i c u l t to see that — = F(x,y) > o f o r the points l y i n g to the r i g h t of the curve F(x,y) = o If = F(x,y) < o f o r the points l y i n g to the l e f t of the curve F(x,y) = o = f(«0 = f ( •*) >o f o r the points below the straight l i n e f (c) = o Co f o r the points above the straight l i n e f (<r») = o The function y ( t ) i s maximum f o r y > o and minimum f o r y < o f o r the points on the straight l i n e f(«*) = o and x ( t ) a t - tains maximum f o r x > o and minimum f o r x < o f o r the points l y i n g on the curve F(x,y) = o, The direction, of motion i s shown i n f i g . 9 . The curve F(x,y) = o, the straight l i n e f (-*>) = o and the axes of co-ordinates divide the plane (x,y) into eight r e - gions. As before we introduce p o l a r co-ordinates x = r cos 4, , y = r s i n Then r = x cos <P r«j> = -x s i n 4» <t>. + y sin* + y cos«fr FIG.9 The signs of r and <P i n different regions are given as (1,5) r > o, *p may be Sfc. o (2,6) r may be ^ o, «* > o (3,7) r < o, (4,8) r may be \ o , 4> may be \ o <*> > o. 52 may b e In the \ o. regions First of these is no m o t i o n w i t h bounded p o l a r is served This i f %(x,y) for < is necessary example see i n order o to i n the we a s s u m e > that r a condition angle «f> ( x , y ) o, we (5) a l l we n e e d in pose regions. (l) and > o and «*» that makes 4> > ensure that there (l,5). regions o Our pur- that in the region (l) (5.18) -^(xjy) Next we r e q u i r e must have > o, that (x,y) «v there be < 0 no Bendixson Let the negative motion M(t) show that i t gion This of (2). (cx - y-axis. crosses ay) the cannot is is and f o r i n to see and enters the region follows. motion region (8) as the after i n in the region - It lies tive - dy = in the 19) region the be) + the ( l ) . We (l) and must We c o n s i d e r 2 that enter rate of now the re- change M(t),i.e., c >(x,y) - a ^(x,y) a«s»(x,y) < (5.20) o ( l ) i n v i e w o f (5.17) a n d (5.18). We c o n s i d e r and cx - we intersecting case ( c x - a y ) = a c x - b e y + c *>i(x,y) - a c x + a d y = y(ad that (5 the easy x-axis as the enter It remain done along motions (5) criterion. a m o t i o n M.(t) half region <° a to the periodic -H - < ¥ according in o the and f i n d first quantities. straight their quadrant Since F(x,y) lines point since = of A, o lies cx - ay = A intersection c, d, d - a (A positive) (— _1_ \ c d-a are a l w a y s be-low the a l l , A \ d^aj posi- straight 53 line f (••) F(x,y) must = o, = o for cross gin since due to even o the increase the region (6) and region region This angle it angle of time shows in is is the positive are satisfied Theorem 5.3. = o, the goes regions x-axis. a n d we h a v e Let _1Z the > o. of this (4), of system (5.l) In t r i v i a l by 1954 is the constructing cx - Then under = o to (5), it and A Thus the origin enters (4), and any m o t i o n w i t h of ori- cannot i n regions L(o,o») (l) ay = the (6) requirement following (5), bounThe (8). and (3) bounded we c a n take Theorem 1.2.7 theorem: conditions (5.2) - (5.5), ; 1, the stable i n and C o n d i t i o n asymptotically G u , Cao hao solution i n line any m o t i o n w i t h line 2_X (5.16), (5.18). (5.19) the regions only occur a l l region to region the Thus the curve go tends (2), <: o . the same r e a s o n . be straight Hence the cannot can the straight holds for r cannot either out (l), angle entering i t of i t there bounded, since half = A intersects curve F(x,y) because that bounded. For the the where or ay because cross (2) (3) - motion M(t) intersecting w i t h bounded p o l a r (7) , w h e r e cx S i m i l a r argument (4). (7). ded p o l a r motion the the and cannot i n the enters Thus line F(x,y) (5.20). A f t e r motion polar curve > <p straight a l l A. the remain with the discussed £l6] the large of § = x h ^ y ) § = x h 2 the ( y ) a Lyapunov f u n c t i o n . trivial the the following + *(y) + f(y) In the next s o l u t i o n x=o=y large. stability of the system two s e c t i o n s we- 54 shall discuss the stability if in h = i the ( y ) Jf = hg(x) Jf = xh^y) Jf = large x a + x + of y by and mostly by qualitative bx methods. ay + + h (x) 2 y 55 6. The stability = h dt (x) x i n the + satisfy us consider the Let the dinates x + ay Jf = h (x) x + by ay, the right (6.1) that 2 hand sides conditions guaranteeing us assume sake of the of the existence system (6.l) and uniqueness of that < o, the of right the y-j ^2 i h ' ~ - (x) — the second, o, of not see and a < (6.1) curves are (y)< o, o, h (x) > 2 on the given y * o, (x,y) o x * o. plane. The a c c o r d i n g as x $ >^o according x h^(y) x + ay origin is the = third as o ' 6 ^o and h ( x ) 2 quadrants only point or- o ( x We by >± o f o u r t h and f i r s t , the to b < o we t a k e the curves and c o n s e q u e n t l y difficult of (yi) '— x ~ shows t h a t i n the * hand sides graphs - x definiteness h dx dt x + 1 solution. represent l i e h , (y) system = ^ ( y ) 2 This the H ah (x) Por = dx by. We a s s u m e 6a. of • dt ^ Let every large of x + by 2 = ) o respectively equilibrium. It is that = h^(y) x + ay < o to the right = h^ (y) x + ay > o to the left of the curve h^(y)x + ay = o clx — of the c u r v e h^ ( y ) x + ay = o 56 |f = h (x)x + by < o to the left |f = h (x)x + by > o to the right 2 2 The f u n c t i o n y ( t ) h (x)x + by on curve 2 the direction hand the of sides curves = o of the curve h^ ( y ) x + ay = o f o r motion and the of (6.l) equal to co-ordinate eight regions. co-ordinates we s e e signs of i n these ^» o; are by x(t) is putting represented o = o curve o a n d maximum f o r obtained by = + by 2 y minimum < o. the The right i n fig.10. axes 'd i v i d e polar and y > zero h (x)x maximum o n t h e > o and minimum f o r y < curves h^(x)x + curve y into r the for and the plane attains of The ci Using that the regions are >^ o, tp> (1.5) r (2.6) r (3.7) r may b e *t^o,. (4.8) r <• o , <p m a y may b e < o, <$> m a y b e The easily be o Tfa <p > o be t r i v i a l shown t o In fact, o o solution asymptotically l e t a V - f u n c t i o n be 2V = is 2 J x FIG.J.O stable 2 the sense of Lyapunov. b}J defined h (x) i n dx - ay 2 o Then V = x = x h (x)(h (y)x 2 2 1 (y) h (x) 2 + ay) • aby 2 - ay (x h ( x ) g < o y * = possibly o + by) o for y = o 57 Obviously,V negative for contain the is y * o b < is o, there criterion of the system constructed L ( o , <*>) We n o w s h o w t h a t We c o n s i d e r with the gions must them s i and y increasing quently leaves diction the > the € we c o n s i d e r trajectories the of the increase W x in (l). the The show t h a t with (i n that the h^(x) x (l), This + + with = time. star-ted the increase our same r e a s o n i n g holds for the (6.l) the quantity bounded. consider this + these we write ay) (l) decreasing and (l) re- by) is time - with region region (bx motion motions x line y-axis. any we assumption. region of half are x of the change that straight large leave of for i n the must rate Lyapunov. entering then,since x=o=y, fact no Por not to Next 2 o to regions or does according the are is except angles o. V = o positive infinitely contradicts motion Por cos<*(h (x) motion + by < there of the these starting ay) y becomes o and hence that (2). r motion curve > region shows region the bounded,since i . e . , any suppose cross In H e r e we * n is the (2,4,6,8). regions angle does not is from a Lyapunov f u n c t i o n . take y (6.1) solutions obvious we Since according w i t h bounded p o l a r angles, us stable motions (l,3>5>7). Let is derivative system no p e r i o d i c This o. the = — [~ 4> the = 1.2.7 the leave of y Theorem bounded p o l a r bounded polar for The i n first regions are Bendixson. appearing zero asymptotically the is function. half-trajectory solution + h^(y) definite and p o s s i b l y a positive trivial Since a positive conse- the motion The contra- and enter (5). Next along the 58 — (bx - ay) = b ( ^ (y)x + ay) - a(h (x)x + 2 by) (6.3) = x(bh^ (y) Consider the straight - a h straight a l l A. Let the curve line A be (6.4) + r e g i o n bounded by bx - must cross straight origin,since <p gion above polar angles are T h e o r e m 6.1. then the If t r i v i a l > as x ^ o (6.4) = A the the fourth + h^(y)x ay + analysis region. shows ah (x)< o for x 2 s o l u t i o n of $ (6.l) ay = of the The + ay = straight the the line motion entering it cannot and cannot S i m i l a r argument a l l for We c o n s i d e r o,since (6.3) o intersects (6.4) quadrant. = o. that b o u n d e d a n d we h a v e xh^(y) line y-axis, ay = A b e c a u s e in this curve straight half h^(y)x curve - o then o in. the curve bx ay negative the line (3). T h e = the ay = A and the region the ay - intersects negative, h^(y)x according line bx The ( x ^ o 2 motions following cross enter holds with this the i n re- bounded theorem; o, b < o and h^(y) < o for is asymptotically stable y i n * the large. We s h a l l n o w d i s c u s s 6b. the case when a h ( x ) 2 > o and the condi- tions b are + (y) satisfied. necessarily h (x) ? > o, < From have x t o, o. b bh.j ( y ) (6.5) < o, it h^(y) F r o m (6.2) ah (x) > o, follows that y # 2 < o for we have x * o, y $ o if ah (x) > o. We a s s u m e 2 (6.5) o we a > must o, o, 59 \ y i y ? - = _ _ bh x h (x) 1 ~i x " (y ) g + ( ^ ^ ) - b x ah M 2^ ^ > o x according as x ( >^o 6 > 6 ) ab Since or and y negative, 2 the are positive two or situated xh^(y) + ay in = the xh (x) + by = o xh + = o (y) 1 f i r s t o is above drant and below i n the note that and the ay third curve third xh^(x) quadrant = b^(y)x + a y •> o to the left dx dt = h^(y)x + ay <o to the right |f = h^(x)x + b y <o to the left |f = h (x)x + b y >o to the right The for y xh (x) 2 > o co-ordinate ection of introduce in r = o for axes motion polar different (1,5) function x(t) and minimum f o r + by may b e y > divide and the is y ^ x o are fc»y) curves o, as <i> + by of the of of is positive plane i n the xh^(y) y < into o. eight The that the f o, = o o = o ay = and The As before of r o (2,6) < <p may be we a n d e*» ^ the dir- follows; < o curve curves signs qua- o = + regions. shown i n f i g . 1 1 . and n o t i c e ay xh^(y) the = + by 2 maximum o n ay + xh (x) curve curve further + by 2 curve We + xh (x) the first xh^(y) curve the that (6.6)). of curve the is o curve the of y(t) are = (because and minimum f o r the and such maximum on t h e 4 o; co-ordinates regions as (6.7) quadrants dx dt 2 according curves 2 are negative o 60 o (3.7) r (4.8) r may be < tegral 4> each there (8) o, o, In and \ curve to the f r o m Theorem origin or (since r motions < o, <*> motions entering regions and enter the rate of the go the f o l - to ^ o the work re- the (3,5) (l), (1,7) or i n the regions of i n - i n the regions the change This regions may b e one origin, started the (4) Erugin's either (8) enter of 2.1 o least equilibrium. and (4) ^ regions going lows gions the at of Other of be point . may be w i l l only f8j > o 4> or quantity (4,8)). regions (3), (2) FIG.il (5) (6). (bx - and To (7) We n o w s h o w must show t h i s leave we that these consider ay),i.e.j d —(bx - ay) The intercepts curves = x(bh^ (y) straight with (6.7) the - line axes i n the straight line bx the positive half y-axis. cross the l i n e bx go the origin,since the increase and (7). the origin The as of - - ay time. motions t _> bx of f i r s t the to alv^x)) ay - ^ ay = A = A (-ve), The the 00 . Thus curve motion M(t) o entering we h a v e the x \ o of regions proved the intersects ay the = o region and (6.8) the (2) + this leave holds for positive r e g i o n bounded entering must (6.8) has xh^(y) because and hence S i m i l a r argument after and hence Consider the (A n e g a t i v e ) < as (A negative) co-ordinates quadrant. = A o according this (6) following and cannot cannot region regions or by with (3), tend to theorem: (5) 61 Theorem If 6.2, (6,l) i s asymptotically conditions > o. let that are (6.5) Then from a < > o 2 We n o w a s s u m e 6c. b ah (x) > x $ o, stable in the either b > o satisfied. i t (6.5) o, h (x) y* = y 9 = - x We c o m p u t e the difference = - x = - 2 for o for Por then large or the $ o. The trivial solution sake > o for of h^(y) + o and the we let definiteness < ordinates y o and ahg(x) of the < curves of (6,5). under conditions h^ ( y ) follows that x the o. in We (6.7) are - x. 1 h,(x) _& h y. - 1 2 y ? ^ a (y ^ o according as x ^ o ^ o according as x £ o between ) + a b h (y ) x — 3 — ' ordinates h (x) -Ar x (6.10) b - ( the ah (x) 6 > follows that p o according as x ^ o ab Prom (6.9) cond and f o u r t h the curve second dx = dt (6.10) and quadrants xhg(x) + by quadrant. h.| ( y ) x it It + = is ay and the o i n the not > o curve fourth d i f f i c u l t to and b e l o w i n the that the curve h^(y) x + of the curve h (x) x to the right |f = h (x) x + by <. o to the left is of the se- above of >o of o lies right + by func'tion x(t) verify = the x The quadrant ay to = h (x) 2 + i n the left |f 2 xh^(y) lie the x+ay o two c u r v e s to = h^(y) |f < the curve h^ ( y ) x 2 the curve h (x) minimum on the curve xh^ (y) 2 + x ay o ay = + by = + by + = ay = = o o 62 for y > xh (x) o + by 2 and the curves The a n d maximum f o r signs (2.6) r (3.7) r (4.8) r Let us gions is < and (2) i t of the into i n 4> eight shown i n (6.7) regions. The fig.12. ^0 the and re- see greatest lower not. function h ^ y ) . otherwise bh^ (y) have c^ ^ i n e q u a l i t y bh^ (y) attainted, curves > o the sign The ^ o or is - strict 1 2 FIG. afc^x) > o h^(y) < ah (x) > 2 admitted i f for - x t o, + o is violated. b o we h a v e b c . j h^(y) y 1 does not inequality holds. - attain Let us ah (x) o. 2 its greatest suppose that then c and i $ Vy) h„(x) * If o. curve o positive the y < plane motion are is equality is > may b e (6) the maximum o n t h e are lower bound otherwise c^ divide >^o, <p o, is i n 4> \ o, <*> compute We t h e n The and and y ( t ) and minimum f o r axes ep m a y b e may b e f i n i t e From o, > o < o, direction of regions > c,| b e bound r may b e whether Let the of different r y co-ordinate and (1.5) = o for y we l e t ^ (y) > < bc - (6.11) b T, a = + * (y), then o (y) < -(b + c 1 ) 63 (6.1l). from P o r h ( x ) we c a n take 2 h (x) = 2 C 1 + 3. » ^ ( x ) , where ^ ( x ) > o Now • r • <*• = = • - x s i n * - + y cos s i n <p ( x h ^ y ) = - sin4>J^x(c 1 * + a y ) + c o s <p ( h ( x ) x + by) 2 + ^ ( y ) ) + ayj + C os*^(^|i + ^(x)) x + by] Then 2 <P = - a s i n * 2 be. + c o s <p + 1 (b - c, ) s i n * a 2 - ^ ( y ) = - 2 a cos * c o s * 1 f , 2 I t a n <p I - s i n 4. b-c, — _ L t a n 4» a cos* bci 1 —rr- 1 a^ J - /• / v (yj s i n * 2 ( * * p - 2 a cos<* ( t a n <P - c 3;) ( t a n 4> + .21) ID - The sign expression only while passing — > - —L a a since b line slope — x with a + c. < ' lies ( t a n * * s c o s <$> multiple of c o s - ^ ( t a n ^ a 2 * + c (6.12) 1 ) c a n change ~ • i t s b 1 t h r o u g h t h e v a l u e s — and - —— The v a l u e C & o and a above > o and below f o r x < o. < o. a. Since — > the straight The s t r a i g h t ~ a 1 a c line with lines , the straight 7 slope - 2l. f o r a y = -rx and y = & l i e i n the second lie i n t h e r e g i o n s ( 2 ) a n d (6). P o r t h i s the straight line ^ ) & ( y ) s i n 4> c o s 4> + ^ (x) 2 o 1 + £ 2( x ) c o s * = + c and f o u r t h y = - ^ 1 _ a x quadrants. lies We s h o w t h a t we h a v e 2L they do n o t o n l y t o show below the curve °2_ xh_(x) ^ that + by = o x 64 in the second this i s h - so, quadrant Thus 2^ ) - 1 x c x x/ 1 r-( that the i n region increase <* h (x)) keeps first In according as x t h e same s i g n quadrant. (2) f o r a l l t i m e . a r e no p e r i o d i c y becomes that - we c o n s i d e r ay = A and in of intersection the second tive. or fourth We c o n s i d e r A(+ve), the positive Since —(bx-ay) lows the region that half remain the motion enters since • > o and the curve bx - ay = q and y of time. origin The m o t i o n or enters reasonings sis shows <. o , the region 1 i n region , - a 1 \ A (b+e )J * T h i s line line l3 or bx - ay = ay + c^x = o . a n d <p > o i n (2), i t above the region the region f o l (3)> the straight (4) w i t h (4) e i t h e r o i n region angle line increase tends to the (4). S i m i l a r (5), ( 6 ) , (7) a n d (8). T h e a b o v e bounded p o l a r -es nega- (3). H e r e , i n t h e r e g i o n (3),since r < any motion with i n the as A i s p o s i t i v e + ay = o l i e s entering hold f o r the regions that quadrant the motion enters after C and the straight the region xh^(y) remain lines bounded b y t h e s t r a i g h t < o i n the second from ay + c^x = o according y-axis point (2) w i t h and then the straight i s given b y ^ ^ j quadrant the L e t a motion M(t) infinite i t cannot o £ i s the only the motion M ( t ) cannot We n o w s h o w t h a t For this motions. / A point > positive. ;,M(fc)must e n t e r t h e r e g i o n otherwise bx Their fact throughout seen to be (6.5)>the o r i g i n ( l ) . The m o t i o n of time, >± o ? to conditions = hj(y) x + ay i t follows the - (2) a n d (6) a n d w h i c h i s e a s i l y equilibrium and there start — = (6.12) According of quadrant. bc — we s e e f r o m regions i n the fourth because — r — + x a n d above i s bounded. analy- For the 65 straight tive line half L(o, y-axis. stable following theorem.* are (6.5) i n the small stable = —— the trivial large that i f the the = x trivial — > o, o s o l u t i o n of i t take solution is and proved asympthe conditions system is posi- (o.l) is asymptotically to Lyapunov). trivial — a. s o l u t i o n be stable i n the example, + m + £ (x), <0 does not contain any constant t e r m a n d h^ ( y ) does not contain any constant term. tion can be written y $ provided we t a k e , f o r *^o( ) we c a n L y a p u n o v , t h e n we h a v e (i.e., according + & of that o or h^(y) i n the realized, x > requirement can be hp( ) b then small The sense either i n Theorem 1.2.7 we a s s u m e the satisfied, asymptotically stable i n If 6.3. appearing Now i f totically Theorem a* ) The = c ^Ax] where m > o and + equations *^(y), of where f i r s t ^ approxima- as dx at The characteristic = ° 1 equation ?» - X is Oj a bc_j + m a ^ - ( b or The roots of this ensures the c.j)A e q u a t i o n have b which + + c„ < ^ + o = o >- b - a m = negative and - s t a b i l i t y i n the am > o real parts since 3 o. s m a l l and which i n turn, (y) ensures 66 the stability i n The A (b = which + the the large. origin is c. ) + 4am motions ^ tend au This - is obtained 4» r = writing called critical i t satisfies u = In the (a, - tan * ical one it is direction integral direction case A , where - in an + au = either enters o expression + m) = • -. T h e direction by) x + two L The question whether decided by using Frommer' , criterion On s i m i l a r If i n b lines = o, the i t then large direction. an i n f i n i t e number stable = along is directions called are singular if o , ordinary critical a finite tically + m) tan * enters 6.4. by • A c r i t i c a l directions given the cos the as equation curve. Theorem o, •1 u ay) according > o r i g i n are + c. otherwise a focus <#» = o directions. the or b) putting 4> (c^x sin and by o. to + by a node is the under of Along number along of ordinary trajectories a particular trajectories easy the or only or c r i t only ordinary one can be [*15j. to trivial prove the solution conditions following of (6.5) (6.l) is theorem; asympto- 67 The s t a b i l i t y 7. Let i n the large us consider We a s s u m e tions We f i r s t Theorem in 7.1. If one o f t h e s e the trivial Proof. the system (y) + = ^ |d fv = bx + h (x) conditions solution of h^Cy)^ 2 (7.l) 7 - l } satisfy of every condi- solution. theorem: o, y * o; h (x)$ o, x # o 2 inequality (7.l) i s asymptotically o f t h e theorem total (at is satisfied), stable i n the least then large. i s time = bx(xh = 2 bx derivative 1 i n view of (y) + ay) (y) - ay 2 = bx h - 2 ay 2 (7.l) i s a y ( b x + hg(x)' y ) 2 ( x ) <o f o r x * o, = o possibly Obviously,V(x,y) function jectory 1.2.5 of anduniqueness the strict 2V(x,y) V h (x)y. We a s s u m e a < o , b > o . A L y a p u n o v f u n c t i o n f o r ( 7 . l ) u n d e r t h e conditions Its sides the following o and ( y hand the existence = bx + ay 2 the right prove ab < = xh^Cy) + a y , J f Jf that which guarantee of ^ (7.1) e x c e p t are satisfied Next i s an i n f i n i t e l y large hL,(y) + h ( x ) < o , 2 the origin. which proves we c o n s i d e r * Hence h ^ y ) hg(x) - positive any p o s i t i v e half theorem. (7.1) under conditions ab>o for definite a l l the conditions t h e above the system o on x = o or y = o and x = o o r y = o does n o t c o n t a i n of y x * o y * o of traTheorem 68 The Theorem Theorem If ab > o. stable in the 7.2. The Theorem Theorem are the proved i n the same w a y as 7.3. stable small. proof of then the t r i v i a l large under Theorem 7.3 solution conditions goes on of is (7.1) asymp- (7.2). similar lines as the proof and c o n d i t i o n s (7.2) 6.3. satisfied, tically can be 4.1. totically of f o l l o w i n g theorem If either then the i n the h^(y) > t r i v i a l large o or h ^ x ) s o l u t i o n of provided i t is > o system (7.l) asymptotically is asympto- stable in 69 8. Remarks. There connection with 8.1. f i r s t The We s h o w e d t h a t the some results question stability (2.2.2) - (2.2.5) fied. are we h a v e arises i n either questions c o n d i t i o n s A and B or A and C are satis- ditions? In the author's view a n d f (o^) as well as the stability (6.1) and struct naturally arises is this : which ditions the i n order that Having the (7.1) large the using does | | = x h |f = f(x) and (7.1) on the right It the is future. the the 1 ( y ) are + ques- form should could be those con- second problem of in necessary boundaries cases where hold. of the trivial two s y s t e m s . the solution of We w e r e Is not i t able systems to possible conto system ay h (x) 2 y p a r t i c u l a r cases hand s i d e s author's + The special conditions q u a l i t a t i v e methods. for what discussed not a Lyapunov f u n c t i o n f o r (6.1) the found t h i s , stability Lyapunov functions construct in unlikely. s t a b i l i t y c o u l d be i n sufficient most We d i s c u s s e d 8.2. of of and is sufficient? and r e g i o n s derive necessary i t have 2.2.1. Theorem to to large with i n addition possible P(x,y) in i f not then answered holds it which be obtained. i n connection the Is tion remaining to of aim to the above under suitable con- system? investigate these questions 70 BIBLIOGRAPHY 1. A i z e r m a n , M . A . , On a p r o b l e m c o n c e r n i n g t h e s t a b i l i t y i n l a r g e of dynamical systems, U s p e h i M a t . Nauk ( N . S . ) 4(1949), n o . 4 ( 2 8 ) a 187 - 1 8 8 . (Russian) 2. Barbashin, in 453 3. the E.A.; N . N . , Krasovskil, large, E . A . , Barbashin, Prikl. P r i k l . case of Mat. Meh. equation On t h e existence asymptotic 6. Ershov, 7. On t h e s t a b i l i t y i n the automatic r e g u l a t i o n , - 72. (Russian) Ershov, B . A . , Prikl. One t h e o r e m Mat. Meh. of the Lya- stability C a r t w r i g h t , M . L . , On s t a b i l i t y o f s o l u t i o n o f f e r e n t i a l equations of the f o u r t h o r d e r , M e c h . 9(1956), 185 - 193. B.A., of i n 18(1954), 345 - 350. 5. of 61 motion 16(1952), 629 - 632. E . A . ; N . N . ,Krasovskil, large, of 86 0952), (N.S.) non l i n e a r Mat. Meh. punov functions i n the the Nauk SSSR On s t a b i l i t y o f third order, (Russian) 4. Akad. stability (Russian) - 456. Barbashin, Dolcl. On t h e the P r i k l . large 18(1954), 381 - 383. certain d i f Quart. J . of ofoucertain Mat. Meh. on s t a b i l i t y of (Russian) system 17(1953)> motion i n the large, (Russian) 8. Erugin, N.P., On c e r t a i n q u e s t i o n s a b o u t t h e s t a b i l i t y o f m o t i o n and q u a l i t a t i v e t h e o r y of d i f f e r e n t i a l e q u a t i o n s i n the large, P r i k l . M a t . M e h . 14(1950), 459 - 512. (Russian) 9. Erugin, N . P . , Q u a l i t a t i v e i n v e s t i g a t i o n of i n t e g r a l curves a system of d i f f e r e n t i a l equations, P r i k l . Mat. Meh. 14(1950), 659 - 664. ( R u s s i a n ) 10. Erugin, of 11. Erugin, N.P., motion, N . P . , equations, 12. Erugin, N.P.. Some general P r i k l . On t h e Prikl. questions Mat. Meh. c o n t i n u a t i o n of Mat. Meh. the theory 228 solutions of of (Russian) Prikl. stability 236. (Russian) differential 15(l95l), 55 - 58. Theorem on i n s t a b i l i t y , 16(1952), 355 - 361. of 15(l95l), of (Russian) Mat. Meh. 71 13. Erugin, N.P., On o n e p r o b l e m o f t h e o r y o f s t a b i l i t y tem of automatic r e g u l a t i o n , P r i k l . Mat. Meh. 620 - 628. (Russian) 14. E r u g i n , N . P . , Methods of b i l i t y i n the l a r g e , (Russian) 15. Frommer, M . , Die integralkurven einer gewohnliehen B i f f e r e n t i a l g l e i c h u n g e r s t e r O r d n u n g i n d e r Umflebung r a t i o n a l e r Unbestimmtheitsstellen, M a t h . A n n . 99(1928), 222 - 272. 16. Gu, Gao-hao. 17. Krasovskil, N . N . , system of of equations 17(1953), 339 - 350. 18. two e q u a t i o n s , (Chinese On s t a b i l i t y three Krasovskil, N . N . , a sys- A . M . Lyapunov and questions of staP r i k l . M a t . M e h . 17(1953), 389 - 400. On s t a b i l i t y 4(1954), 347 - 357. of 16(1952), of Acta. Math. with Russian Summary) solutions one i n the of large, Prikl. non Sinica linear Mat. Meh. (Russian) On s t a b i l i t y d i f f e r e n t i a l equations, 672. (Russian) of solutions P r i k l . of Mat. Meh. systems of two 17(1953), 651 - 19. K r a s o v s k i l , N . N . , On s t a b i l i t y i n t h e l a r g e acting disturbances, P r i k l . Mat. Meh. (Russian) 20. K r a s o v s k i l , N . N . , On t h e r e v e r s i b i l i t y o f t h e o r e m o f K . P . P e r s i d s k i i on u n i f o r m s t a b i l i t y , P r i k l . M a t . M e h . 19(1955), 273 - 278. (Russian) 21. Malkin, I . G . , On one p r o b l e m o f t h e o r y o f s t a b i l i t y tem of a u t o m a t i c r e g u l a t i o n , P r i k l . Mat. Meh. 365 - 368. (Russian) 22. 23. under constantly 18(1954), 95 - 102. of a sys- 16(1952), iit«. fw« Malkin, I.G., 0n question of r e v e r s i b i l i t y of^theorem of L y a p u n o v a b o u t a s y m p t o t i c s t a b i l i t y , P r i k l . M a t . M e h . 18(1954), 129 - 138. (Russian) A Massera, J.L., of Math. On L i a p o u n o f f ' s conditions of stability, Ann. (3) 50(1949), 705 - 721. 24. Ogorcov, A . I . , On s t a b i l i t y o f s o l u t i o n s o f t w o n o n - l i n e a r d i f f e r e n t i a l e q u a t i o n s o f the t h i r d and f o u r t h orders; P r i k l . M a t . M e h . 23(1959), 179 - 1 8 1 . (Russian) 25. Pliss, V . A . , large, Qualitative Prikl. picture Mat. Meh. of integral curves 17(1953), 541 - 554. in the (Russian) 72 P l i s s , V . A . , Necessary and s u f f i c i e n t c o n d i t i o n s of s t a b i l i t y i n the l a r g e f o r a system of n - d i f f e r e n t i a l equations, D o k l . A k a d . N a u k S S S R ( N . S . ) 103(1955), 17 - 1 8 . (Russian) P l i s s , V . A . , I n v e s t i g a t i o n 03 .a,; n o n - l i n e a r d i f f e r e n t i a l e q u a t i o n of the t h i r d o r d e r , D o k l . A k a d . N a u k SSSR (N.S.) 111(1956), 1178 - 1 1 8 0 . (Russian) Pliss, V . A . , large, Simanov, S.N., order, Wintner, Certain problems On s t a b i l i t y Prikl. A . , The ferential Zubov, V.I., of theory of stability L e n i n g r a d U n i v e r s i t y P u b l i s h i n g House of Mat. Men. non l o c a l equations, non stability, of existence Amer. general Prikl. equation 17(1953), 369 - 372. p.YobXeTOsWffittheory construction linear J. of ^second i n of the (Russian) of the third (Russian) ordinary dif- 67(1945), 277 - 284. Math. solution Mat. Meh. problem i n 1958. method the of region Lyapunov of 19(l955), 179 - 210. and asymptotic (Russian)
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Stability in the large of autonomous systems of two differential equations Mufti, Izhar-Ul Haq 1960
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Title | Stability in the large of autonomous systems of two differential equations |
Creator |
Mufti, Izhar-Ul Haq |
Publisher | University of British Columbia |
Date Issued | 1960 |
Description | The object of this dissertation is to discuss the stability in the large of the trivial solution for systems of two differential equations using qualitative methods (of course in combination with the construction of Lyapunov function). The right hand sides of these systems do not contain the time t explicitly. First of all we discuss (Sec. 2.) the system of the type [equations omitted] These equations occur in automatic regulation. Using qualitative methods we determine sufficient conditions in order that the trivial solution of system (l) be asymptotically stable in the large. In this connection we note that a theorem proved by Aĭzerman for the systems of two equations (Sec. 3), namely, for the systems [equations omitted] In the case of system (2) we give a new proof of a theorem which asserts that if c² + ab ≠ o, then the trivial solution is asymptotically stable in the large under the generalized Hurwitz conditions. The theorem was first proved by Erugin [8]. For system (3) Malkin showed that the trivial solution is asymptotically stable in the large under the conditions a + c < o, (acy - bf(y)) y > o for y ≠ o and [formula omitted] We prove a similar theorem without the requirement of [formula omitted] We also discuss (Sec. 4) the stability in the large of the systems [equations omitted] We consider (Sec. 5) again the system of the type (l) but under assumptions as indicated by Ershov [6] who has discussed various cases where the asymptotic stability in the large holds. Not agreeing fully with the proofs of these theorems we give our own proofs. Finally we discuss (Sec. 6 and 7) the stability in the large of the systems [equations omitted] under suitable assumptions. As a sample case we prove that if ab > o,then the trivial solution of system (4) is asymptotically stable in the large under conditions h₁(y) + h₂(x) < o , h₁(y) h₂(x) - ab > o, for x ≠ o, y ≠ o |
Subject |
Mathematics Differential equations |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-12-16 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080607 |
URI | http://hdl.handle.net/2429/39768 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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