The purpose of this note is to extend Liapunov's second method to the case of differential inclusions, when viability requirements are made and when the Liapunov functions are continuous

Aubin, J.-P.

International Institute for Applied Systems Analysis (IIASA)

A Viability Approach to Liapunov's Second MethodAubin, J.-P.IIASA Working PaperWP-84-067September 1984 Aubin, J.-P. (1984) A Viability Approach to Liapunov's Second Method. IIASA Working Paper. WP-84-067 Copyright © 1984 by the author(s). http://pure.iiasa.ac.at/2451/ Working Papers on work of the International Institute for Applied Systems Analysis receive only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute, its National Member Organizations, or other organizations supporting the work. All rights reserved. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage. All copies must bear this notice and the full citation on the first page. For other purposes, to republish, to post on servers or to redistribute to lists, permission must be sought by contacting repository@iiasa.ac.at NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR A VIABILITY APPROACH TO LIAPUNOV'S SECOND METHOD Jean-Pierre Aubin September 1984 WP-84-67 Working Papers are interim reports on work of the International Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily repre- sent those of the Institute or of its National Member Organizations. INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, Austria PREFACE In this paper, viability theorems are used to provide short proofs of extensions of Liapunov's second method to the case in which differential equations are replaced by differential inclusions, the Liapunov functions are only required to be continuous and viability constraints are present. This research was conducted within the framework of the Dynamics of Macrosystems study in the System and Decision Sciences Program. ANDRZEJ WIERZBICKI Chairman System and Decision Sciences Program ABSTRACT The purpose of this note is to extend Liapunovls second method to the case of differential inclusions, when viability requirements are made and when the Liapunov functions are con- tinuous. A VIABILITY APPROACH TO LIAPUNOV'S SECOND METHOD Jean-Pierre Aubin When f is a continuous single-valued map from an open sub- set R of lRn to lRn and V is a differentiable function defined on R, the Liapunov method derives from estimates of the form informations on the behavior of a solution x(*) to the differen- tial equation x' = f(x), x(0) = xo given by inequalities of the form where w is a solution to the differential equation (see for instance ~oshizawa [ 19661 ) . We shall extend this result when we replace the differential equation by a differential inclusion, when we require viability conditions and when we assume that V is only continuous (because "interestingw examples of functions V are derived from non- differentiable norms, for instance). We look for solutions x(*) to ( 4 )  for almost all tE[ 0.~1 ,x' (t) E F(x(t) ,x(O) = xo given in K satisfying i) Vt E [ O,T] ,x (t) belongs to a closed subset K (viability) (5) I ii) Vt E 0,Tl ,V(x (t) 5 w (t) where w(t) is a solution to the differential equation (3) . For that purpose, we choose among the concepts of tangent cones to subsets and generalized directional derivatives of a function - the contingent cone TK(x) to K at x, defined by (6) T~ (x) : = t v E nn 1 liminf d(x+hv,K) = 0) h + O +  h introduced by Bouligand [1932], (see also Aubin-Cellina [I9841 section 4.2, pp.176-177). - the hypo-contingent derivative D-V(x) of V at x, defined by (see Aubin-Cellina [1984], section 6.1, p.287). We shall prove the following: Theorem 1: Let V be a nonnegative continuous function defined on a neighborhood of the closed subset K and I) be a nonpositive con- tinuous function from JR+ to JR satisfying $(O) = 0. Let xo € K be given. (a) We assume that (8) F is upper semicontinuous with non-empty compact convex values. 4 If we replace estimate (1) by (9) Vx E K , ~ v  E F (x) fl TK (x) such that D-V(x) (v) - < ( V ( x )  ) there exist T > 0 and solutions w ( * )  , x ( * )  to the problem (31, (4) and (5). (b) We assume that (10 F is continuous with non-empty compact values If we posit the stronger estimate Vx E KtF(x) C TK(x) and sup D-V(x) (v) 5 $(v(x)) vEF (x) there exist T > 0 and solutions w(*), x(*) to the problem ( 3 )  (4) and (5). (c) We assume that F is Lipschitz on a neighborhood of K and has non- (12) empty compact values and $ is Lipschitz on a neigh- borhood of [O,wo1 Then estimate (11) implies the existence of T > 0 such that any solution (w(*) ,x(*)) to ( 3 )  and (4) satisfies property (5) . Remark : If we assume furthermore that F is bounded, we can take T = + in the above theorem. This implies that w(t) converges to some w, when t - where w,E[O,~(x~)l is a solution to the equation I) (w,) = 0. If I) (w) < 0 for all w > 0, we then deduce that (13) lim V(x(t)) = 0. t - t W  Proof of Theorem 1: (a) We set: We introduce the viability domain which is a closed subset of IRn x IR x IR (where wo > V (xo) ) . We observe that if v E TK (x) satisfies D-V (x) (v) 2 q (~(x) ) then (v,q(w),O) belongs to ~ ~ ( x t w t h )  Indeed, since v belongs to TK(x), there exist sequences of elements hn > 0 and vn converging to 0 and v such that Vn, x + hnvn E K By the very definition of D-V(x)(v), there exists a sequence of elements an E IR converging to D_V(X) (v) such that, for all n 1 0, If $(V(x)) = w, we take bn:= an + $(V(x)) - D-V(x) (w) if D-V(x) (w) > - and bn = $ (V (x) if D - V (x) (w) = - Q). If $ (V (x) ) < w, we take bn:= $ (w) and we deduce that V(x+hnvn) 2 w + hn$(w) for n large enough because V is continuous. In summary, bn converges to w and satisfies Vn,x + hnvn E K and V(x+hnvn) 2 w + hnbn This shows that (x+hnvn,w+hnbn,A+hnO) belongs to K and thus, that (y$ (w) , 0) belongs to TK (x,w, A). We consider now trajectories x ( * )  ,w ( a )  ,A ( * )  of the differ- ential inclusion [ i) (XI (t) ,w! (t) ,A' (t) ) E G(x (t) tw(t) A (t) (19) 1 ii) 0 w 0  A 0 1 = (xotV(x0) 10) which are viable in the sense that (20) fit E [OrTI r (x(t) rw(t) rA(t)) E K We then observe that A(t) = 0, that x(*) is a solution to (4), that w is a solution to (3) and that (20) implies properties (5). (b) If F satisfies assumptions (8) and (9), then G is also upper semicontinuous with compact convex values and G(x,w,A) nTK(xlw,A) # g. Hence Haddad's viability theorem (see Haddad 119811 Aubin-Cellina [1984] , Theorem 4.2.1, p.180) implies the existence of a solution to (19)-(20) on some interval. (c) If F satisfies assumptions (10) and (ll), then G is contin- uous with compact values and G (x,w, A) C TK (x,w, A) . Hence the viability theorem of Aubin-Clarke [1977] (see also Aubin-Cellina [1984], Theorem 4.6.1, p.198) implies the existence of a solution to (19)-(20) on some interval. (d) If F satisfies assumptions (11) and (12), then G is Lipschitz with compact values on a neighborhood of K and G(x,w,A) c TK (x,w,A) . Hence the invariance theorem of Clarke [19751 (see also Aubin-Cellina [1984], Theorem 4.6.2, p.202) shows that any solution of (19) satisfies (20). Remark : We can solve in the same way the case when we consider I i) p nonnegative continuous functions Vi defined .J on a neighborhood of K ii) p nonpositive continuous functions I); from J I IR, to IR+ satisfying $ .  (0) = 0 3 and when we replace condition (5) by i) fit E 10 ,TI ,x (t) E K (22) ii) fit E [O.TI ,V = 1,. . . ,p,Vj (~(t) 5 W .  (t) j J where w.(*) is some solution to the differential equation 3 We have to replace Liapunov estimates (9) by Vx E K,3v E F(x) n TK(x) such that, V = 1 ,..., p, D-V.(x) (v) 2 qj(V (XI) j I j and estimate (11) by Vx E K,F(x) C TK(x) and V = l,...,p, j I sup D V (XI (v) 5 $ (Vs (XI vEF (x) - 3 3 3 Therefore, the asymptotic properties of solutions to the differ- ential inclusions (4) are concealed in the following function defined by for set-valued maps F satisfying (8) or the function $ defined by 1 (27) lJ1(w):= sup SUP D-V(x) (u) V(X)=W uEF(x) for set-valued maps F satisfying (10). Hence any continuous function $ larger than $ (or ql) will 0 provide solutions w ( * )  to (3) estimating the value V(x (t) ) on some trajectory of the differential inclusion (4). For instance, we obtain the following consequence on asymp- totic stability. Corollarv: Let V be a nonnegative continuous function defined on a neighborhood of K and let xo be given. Let F satisfy assumption (8). We assume further that A. EIR achieves the finite maximum in . - Po . - sup in£ (hw-$, (w) ) A Q R  wz0 If po > 0 and V(xo) 5 gg (1-e-'oT) there exists a solution x(*) to the differential incyusion (4) satisfying - po(t-T) if ho = 0 If po 5 0 and ho < 0, then there exists a solution x(*) to the differential inclusion (4) satisfying 1 (30) vt 2 0, V(x(t)) < (po - c ehot) where co = po-hoV(xo) 0 0 Proof: We take $(w) := how - po Remark : Theorem 1 implies directly the asymptotic properties on U- monotone maps as they appear in corollaries 6 . 5 . 1  and 6 .5 .2 ,  pp. 320- 321 of Aubin-Cellina (19841. Let U : IRn x IRn +IR+ (+ Urn) be a nonnegative function satis- f ying (31) U(yty) = 0 for all y E K which plays the role of a semidistance (without having to obey the triangle inequality). We assume that for all x E Kt x + U(x,y) is locally Lipschitz around K and we set Let @ be a continuous map fromlR+ tom+ such that @(O) = 0. We say that F is U-monotone (with respect to @ )  if Let us assume that c E K is an equilibrium of F (a solution to 0 E F(c)) and that -F is U-monotone with respect to $. Then we observe that by taking V (x) := U (x,c) , we have Let w(*) be a solution to the differential equation If F satisfies either (8) or (lo), there exists a solution to the differential inclusion (4) satisfying (36) U(x(t),c) f ~ ( t )  for all t E [O,T]. REFERENCES Aubin, J-PI and A. Cellina [I9841 Differential inclusions, Springer Verlag, Berlin. Aubin, J-PI and F.H. Clarke [I9771 Monotone invariant solutions to differential inclusions, J. London Math. Soc. 16, 357, 366. Bouligand, G. 119321 Introduction 2 la gzometrie infinitgsimale directe Gauthier-Villars, Paris. Clarke, F.H. [1975] Generalized gradients and applications. Trans.Am. Math.Soc. 205, 247-262. Haddad, G. [I9811 Monotone trajectories of differential inclusions and functional differential inclusions with memory. Israel J.Math. 39, 83-100. Yoshizawa, T. [I9661 Stability theory by Liapunov's second method. The Mathematical Society of Japan, Tokyo. 

A Viability Approach to Liapunov's Second Method

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