Papageorgiou, Nikolaos S.

English

Institute of Mathematics AS CR, v. v. i.

Archivum MathematicumNikolaos S. PapageorgiouOn the existence of ψ-minimal viable solutions for a class of differential inclusionsArchivum Mathematicum, Vol. 27 (1991), No. 3-4, 175--182Persistent URL: http://dml.cz/dmlcz/107419Terms of use:© Masaryk University, 1991Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czARCHIVŮM MATHEMATICUM (BRNO) Tomus 27 (1991), 175 - 182 ON THE EXISTENCE OF V-MINUVfAL VIABLE SOLUTIONS FOR A CLASS OF DIFFERENTIAL INCLUSIONS NlKOLAS S. PAPAGEORGIOU (Received September 22, 1988) ABSTRACT. In this paper we establish the existence of ^minimal viable solutions for a class of differential inclusions with a Hausdorff continuous orientor field defined on a general Banach space and satisfying a compactness hypothesis and a strong Nagumo type condition (theorem 3.1 and 3.2). When the space is finite dimensional, we show that the strong Nagumo condition can be weakened to a regular Nagumo type (tangential) condition. 1. INTRODUCTION In a recent paper Falcone-Saint Pierre [6] established sufficient conditions for the existence of slow viable solutions for a class of differential inclusions defined on a finite dimensional Banach space. In this note we generalize the results of Falcone-Sait Pierre [6], by relaxing some of their hypotheses and by proving an existence result for finite dimensional differential inclusions. Consider the following multivalued Cauchy problem on a Banach space X: x(t) Є F(x(t)) a.e. *(0) = x0£KÇX [x(t)єK, tЄT=[0,b)} (*) In their works, Deimling [4] and the author [13], proved that under some com­pactness type hypothesis on the orientor field F(x), a necessary and sufficient condition for existence of solutions of (*), is that for all * € K> F(x) HTK(X) £ I (Nagumo type condition). Here TK(X) denotes the Bouligand tangent cone to K at x. 1991 Mathematics Subject Classification: 34A40. Key words and phrases: differential inclusions, measure of noncompactness, Hausdorff con-tinuity, lower semicontinuity, Bouligad cone, Arzela-Ascoli theorem, ^-minimal solution, Kura-towski convergence. 176. NIKOLAS S. PAPAGEORGIOU In this paper we will be looking for a special type of viable solutions, namely solutions with velocity which is minimal with respect to a certain criterion if>(-) (̂ (•J-minimal solutions). So let %l>: X —• E be a continuous, convex function. We say that a trajectory x(-) of (*) is "ifr-mimmal, viable" if and only if ť(x(<)) = inf{V(*) : * € F(x(t))C\TK.{x(i))} a.e. Note that when i/>(x) = ||x|| (the norm function), we recover the notion of slow solution, which is important in mathematical economics and control theory (see Aubin [1] and Henry [7]). In this case V>(-) is nothing else but the metric projection on the set R(x) = F(x) C\ TK(x). Recall that if the underlying state space X is a strictly convex, reflexive Banach space and K C X is nonempty, closed, convex, then the metric projection function x —• proj(ar, K) is single valued. 2. PRELIMINARIES Let X be a Banach space. Throught this paper we will be using the following notations: Pf(e)(X) = {A C X : nonempty, closed, (convex)} and Pjt(c)(X) = {A C X : nonempty, compact, (convex)}. On Pf(X) we can define a generalized metric /i(», •), known as the Hausdorff metric, by setting h(A,B) = max{sup(inf \\a - 6|| : 6 € -B), sup(inf | | 6 - a\\: a G A)}. a€-4 beB Recall that (Pf(X),h) is a complete metric space. A multifunction F : X —• Pf(X) *s sa-d to be Hausdorff continuous (h-continuous), if it is continuous as a function from X into the metric space (Pf(X), h). More generally if Y> Z are Hausdorff topological spaces, a multifunction F : Y —» 2Z \ {0} is said to be lower semicontinuous (l.s.c), if for all U C Z open, F~(U) = {y 6 Y : F(y) O U ^ 0} is open in Y. If Y, Z are metric spaces, this definition is equivalent to saying that for any yn —• y in Y, we have F(y) C limF(y„) = { :€ Z : \imd(ztF(yn)) = 0}, where d(z,F(yn)) = inf{||z - z'\\ : z' € F(yn)}. We will say that F : Y —• 2Z \ {0} is upper semicontinuous (u.s.c), if for all U C Z open F+(U) ^{yeY : F(y) CU}is open in Y (see Delahaye-Denel [5]). Now let us turn to X being a Banach space, let K C X be nonempty and let x 6 K. The "Bouligand or contingent cone" to K at x is defined by: aw-là-^jfŕS-Ѓ-tì-ь} where for any z 6 X, dK(z) = inf{||* - x'\\ : x1 G K} (see Aubin-Cellina [2]). It is clear that this cone is closed, TK(x) = Tg(x) and if x £ int K> then TK(x) = X. ^MINIMAL VIABLE SOLUTIONS 177 Note that unfortunately TK(X) in general is not convex. Also note that int K £ 0, then for all z G K, int TK(x) ^ 0 (see Aubin-Ekeland [3], p.169). By a ( ) . We will denote the "Kuratowski measure of noncompactness" which is defined on the nonempty, bounded subsets of X. So if .A is such a set we have: m a(A) = inf{d > 0 : A C [J Ak> for some m and A'k8 s.t. diam(Ajfe) < d). Finally given a multifunction F : Y - • 2Z \ {0}, by "graph of F " we will mean the set GrF = {(y, z) 6 Y x Z : z G F(y)}. 3. MAIN RESULTS Let X be a Banach space and V> • X —• E a continuous, convex function. We will be looking for ^--ninimal viable trajectories of (*). Recall that x : T = [0,6] —• X is a "V'-tt-ini-™- viable trajectory", if there exists / G 5'F(X()) — {9 ^ ^ ( ^ O : g(t) G F(x(<)) a e ) 8.t. x(t) = x0 + /0fy(s)ds for all t € T, *(<) G # and tP(x(t)) = inf {^(*) : 2 G F(z(*)) n !*(*(<))} a.e. In our first theorem we will establish the existence of such solutions for a large class of infinite dimensional differential inclusions. For this we will need the fol-lowing simple lemma. Lemma. IfYtZ are HausdorfF topological spaces, F : Y —• 2Z \ {0} is i.s.c, G : y - • 2Z \ {0} .has an open graph and for all y eY, F(y) D G(y) ^ 0, then y — L(y) = F(y) D G(y) is l.s.c. Proof. We need to show that given V C Z open, L~(V) = {y £Y : L(y) 0 V £ 0} = {y € y : F(y)CiG(y)nV ? 0} is open. Let y G L~(V) and z G F(y)f)G(y)nV. Then (y,z) G G r G n (Y x V). Note that since by hypothesis G ( ) has an open graph, GrG O (Y x V) is open. So we can find U\(y) an open neighborhood of y arid W\(z) an open neighborhood of z.s.t. U\(y) x Wi(*) C GrG f)(Y x V). Observe that F(y) H W\(z) £ 0 since it contains z and because by hypothesis F ( ) is l.s.c, we can find U2(y) and open neighborhood of y s.t. F(]f)eW$z) £ 0 for all t/ G U2(y). Set C/(y) = U\(y) H U2(y). Then for all y> G I/(y) we have F(j/) fl Wi(z) ^ 0 while [/(y) x W\(z) C GrG fl ( y x V). So for all y> G U(y), ^ ( j / ) H G(j/) fl V = 1(2/) H V ? 0 =» I-(V r) is open => L() is l.s.c. • Now we are ready for the theorem establishing the existence of V'-minimal viable solutions for (*). Our result extends theorem 4.1 of Falcone-Saint Pierre [6], since our state space is infinite dimensional, the growth hypothesis on the orientor field F() is more general and V() need not be inf-compact ad in [6]. Note that this last fact is very important, because it allows j>(-) to be the norm of an infinite dimensional Banach space and so our existence theorem incorporates the results on the existence of slow solutions (see Aubin-Cellina [2]). 178 NIKOLAS S. PAPAGBORGIOU Theorem 3.1. If K G Pfc(X) with int K ^ 0, V : X - • K is continuous, convex and F : K —> Pjc(X) is a multifunction s.t. (1) F() is h-continouous, (2)\F(x)\<c(l + \\x\$,c>0, (3) a(F(B)) < ka(B) for all B C K nonempty bounded, k>0, (4) F(x) O int TK(x) # 0 for all xeK, then (*) admits a tp-minimal viable solution x(-). Proof. Let G : K -» 2X be defined by G(x) = {t, G Y : tl>(y) < m{(t/>(z) : z G R(x)) = A(x)} where R(x) = F(x) H TK(X). Since F( ) is A-continuous (hence l.s.c. too) and x —• int TK(X) has an open graph (see Aubin-Ekeland [3], proposition 7, p. 169), from the lemma we deduce that x —• F(x) H int TK(X) l.s.c. Hence x - • F(x) fl int Tfr(x) = F(x)f)TK(x) = _R(x) is l.s.c. (see Klein-Thompson [9], propo-sition 7.3.3, p. 85). Also since #(•) is compact valued (.F(0 being compact val-ued because of hypothesis (3)), there exists a z G R(x) (depending on x) s.t. \l>(z) = A(x) =-> G(x) ^ 0 and in fact, since V(') ls a- s o convex, it is easy to see that G(x) G Pkc(X). We claim that G(-) has closed graph. To this end let (xn, yn) G GrG (xn, yn) --* (x,y) in if x X. We have ^O/n) < A(x„) for all n > 1. Since /?(•) is Ls.c. from theorem 4, p. 51, in Aubin-Cellina [2], we have that A(-) is u.s.c. So by passing to the limit we get: limip(yn) = tl>(y) < fimA(x„) < A(x) =>(x,y)€GrG ==-> GrG is closed in KxL. Invoking theorem 1, p. 41 of Aubin-Cellina [2], we get x -* L(x) = .F(x)flG(x) is u.s.c. Also because of hypothesis (4), L(X)DTK(X) ^ 0 for all x G K. Furthermore we have |L(x)| = sup{||z|| : z G L(x)} < \F(x)\ = sup{||z'|| : z1 G F(x)} < c(\ + ||x||) (hypothesis (2)), while for B C K nonempty bounded, since the Kuratowski measure of noncompactness is monotone, we have a(L(B)) < a(F(B)) < ka(B). So if we consider the following viability problem i(ť) Є L{x{t)) a.e. x(0) = «o Є K x(t)€K,teT=[0,b)ì (*)' we see that all the hypotheses of theorem 1 of Deimling [4] are satisfied and so according to that theorem there exists solution x(-) of (*)'. It is easy to see that x(') is the desired V>()-^Binial viable solution of (*). D We can also have an integral selection criterion. "Sen as before let i/>: X —• M be a continuous , convex function and set I^(v) = f* 1>(v(t)) dtf for all t>() € Ll(X) if the integral exists, permitting ±oo. We say thai ^MINIMAL VIABLE SOLUTIONS 179 a trajectory ar() of (*) is "/̂ -minimal viable'' if and only if .fy(i) = inf{/^(v) : v € SVo)}* where R(x) = F(x)nTK(x) and S1^^ = {g G I1**) : *(') G R(x(t))} a.e. Our existence result concerning 7,/r minimal viable trajectories of (*), reads as follows: Theorem 3.2. If X is a separable Banach space, K G Pfc(X) with int K £ 0, tl> : K —* R is continuous, convex, for all z(-) viable trajectories of (*) and aii v G -5H(*())» I*i>(v) *s defined and finite for at least one such v and F : K —• Pjc(X) is a multifunction satisfying hypothesis (1) -+ (4) of theorem 3.1, then (*) admits an I^-minimal viable trajectory. Proof. From theorem 3.1 we know that there exists ^-minimal viable trajectory x(.) of (*). So i>(x(t)) = inf{V>(z) : z G R(x(t))} a.e. Note that R() being l.s.c. is measurable. So we can apply theorem 2.2 of Hiai-Umegaki [8] and get that: inf{/^(V) : v G 5^ ( . } ) } « J inf{V(z) : z G R(x(t))} dt = / il>(x(t))dt = Ii,(x), Jo ==> x(-) is an I^-minimal viable trajectory of (*). D If the underlying state space X is finite dimensional, then we can improve theorem 3.1 by replacing hypothesis (4), with a standard Nagumo type hypothesis. So we have the following existence result. Theorem 3.3. If dim X < oo, K G Pfc(X) with int K £ 0, ^ : X -* Ik is a continuous, strictly convex, inf-compact function and F : K —• Pje(X) is a multifunction s.t. (1) F(-) is h-continuous, (2)\F(x)\<c(l + \\x\\), c > 0 , (3) F(x) fl Tx(x) ^ 0 for all x G K, then (*) admits a ^-minimal viable trajectory x(*). Proof. Let Fn(x) = F(x) + ^B\, where B\ is the closed unit ball iti X. Clearly Fn() is ii-continuous, |Fn(x)| < c(+£) + c||ar|| and Fn(x) f\ int TK(X) £ 0 for all x G K. Consider the following approximating viability problems: ř *»(<) € M*n(t)) a.e. xn(0) = zo € А-[xn(t)£K, ÍGT-=[0,b]J (*)» From theorem 3.1 (note that hypothesis ($) of that theorem is automatically satisfied with k = 0 because of the finite dimensional^ 180 NIKOLAS S. PAPAGEORGIOU every n > 1, (*)n admits a ^-minimal viable solution xn(). Then for all n > 1 we have: IM0II<(«+1) + C||«n(<)ll ae-=> IMOII < IHI + (c + 1)6 + / ' e|M«)|| da. Jo So from Gronwall's inequality we get that for all n > 1 and all t G T IMOII < (IMI + (c+ l)6)exp(c6) = M. Thus ||xn(OII < (c + 1) + cM = M a.e. Therefore {in()}n>i is uniformly integrable in Ll(X) and so {xn(-)}n>i is equicontinuous in C(T,X). It is also bounded. So from the Arzela-Ascoti theorem, we deduce that {xn}n>1 is compact in C(T,X). Hence by passing to a subsequence if necessary, we may assume that xn ->xinC(r,X). Note that Fn(x) —> F(x) (convergence in the sense of Kuratowski [10], p. 339). Because int TK(X) ^ 0, from lemma 1.4 of Mosco [11], we have that Fn(x) H TK(x) = Rn(x) £ F(x) fl TK(x) = R(x) for all x G K. Now we claim that the minmization problem min{<p(z) : z G R(x)} is Tihonov well-posed i.e. it admits a unique solution zn £ R(x) and every minimizing se-quence converges to it. That a solution exists, follows from the continuity of ^() and the compactness of R(x). That is unique, is a consequence of the strict con-vexity of $(•). Finally let {zn}n>i be a minimizing sequence i.e. i)(zn) { X(x) where A(x) is the value of the minimization problem. Without any loss of gen-erality, we may assume that for all n > 1, t/>(zn) < X(x) + 1. Since ip(-) is inf-compact, {zn}n>i is relatively compact and so we may assume that zn —> z. Then V'(̂ n) —• i>(z) = ^(*) ie. z is the unique solution of the minimization problem. Therefore rmn{ip(z) : z G R(x)} is Tihonov well-posed. Without any loss of gen-erality assume V>(0) = 0. Set An(x) = min{il>(z) : z G Rn(x)} and A(x) = min{t/>(z) : z G R(x)}. Since Rn(x) -+ R(x) and the limit problem is Tihonov well-posed, we can apply theorem 3 of Zolezzi [14] and get that An(x) | A(x). Now note that for every n > 1, we have An(xn(<)) < A(xn(t)). Recall that A() is u.s.c. So we get: Im-An(xn(*)) < HSlA(xn(t)) < A(x(t)). Also from the Dunford-Pettis compactness criterion and by passing to a subse-quence if necessary we may assume that xn «-+ i in Ll(X). Then for all A C T Lebesgue measurable we have XAXn 2* XAX in LX(X). Recalling that /^(-) is ^MINIMAL VIABLE SOLUTIONS 181 weakly l.s.c. we get: / V>(XA(V)X(.)) dt < lim / i>(XA(t)in(t)) <ft < 1m. / il>(in(t)) dt = JO .!0 J A = lim I An(*n(<)) dt < J iimAn(*n(0) dt (Fatou's lemma) < <J$x(t))dt, =>Jj(i(t))dt<J^\(x(t)) => VOW) < A(*(f)) a.e. Next we claim that Fn(xn(t)) —• F(x(t)) as n —> oo. To this end, note that for every n > 1, * -* un(x) = n(Fn(x), F(x)) is continuous and «„(*) J 0. So from Dini's theorem we have u„(*) —• 0 uniformly on compacta. Then note that: h(Fn(Xn(t)), F(x(t))) < h(Fn(Xn(t))> F(xn(t))) + h(F(xn(t)), F(x(t))) = Un(Xn(t)) + h(F(xn(t)),F(x(t))) We see that un(xn(t)) —• 0, while from hypothesis (1) we have h(F(xn(t)), F(x(t))) -4 0 as n - • oo => Fn(xn(t)) -^ -?(*(*)) as n -+ oo. Now observe that in(t) e Fn(xn(t)) a.e. and a?n —• x in C(T,X), while i n ^ i in -^(X). Invoking theorem 1, p. 60 of Aubin-Cellina [2] (see also theorem 3.1 0^12]), we get: x(t) € F(x(t)) a.e. Furthermore x(t) G K for all t € 7\ Then for A > 0 we have: dK(x(t) + \x(t)) _ <**(*(< +A)-Ae(A)) Ac(A) . A "~ A S ~ - € { A ) where e($ —• 0 as A —• 0 + . So we have: MdK(x(t) + Xx(t))=0 ^ Aio A => i(t) € rjc(«(l)) a.e. => x(t) e F(*(<)) n TK(x(t)) = £(*(*)) a.e. Because of the uniqueness of the solution of mm{^(z) : z € R(x(t))} = A(s(<)) and since as we saw above V'OKO) £ A(*(0) a.e., we conclude that -c(-) is the desired ^-minimal viable trajectory of (*). O 182 NIKOLAS S. PAPAGEORGIOU If in theorem 3.3, X is strictly convex and tp(z) = ||^||, then the result applies and we get slow viable solutions for (*). By the way, note that there is a minor inaccuracy in the work of Falcone-Saint Pierre [6]. The state space X has to be strictly convex, or otherwise the metric projection need not be single valued. Acknowledgement. The author wishes to express his gratitude to the referee for his (her) corrections and remarks that improved the content of this paper. REFERENCES [I] J.P.Aubin, Slow and heavy viable trajectories of controlled problems. Smooth viability domain In Multifunction and Integrals, ed. G. Salinetti, Lecture Notes in Math., vol. 1091, Springer, Berlin (1984), 105-116. -{2] J.P. Aubin - A. Cellina, Differential Inclusions, Springer, Berlin (1984). [3] J.P. Aubin - I. Ekeland, Applied Analysis, Wiley, New York (1984). [4] K. Derailing, Multivalued differential equations on closed sets, Diff. and Integral Equations 1 (1988), 23-30. [5] J.P. Delahaye - J. Denel, The continuities of the point to set maps, definitions and equiva-lences, Math. Progr. Study 10 (1979), 8-12. [6] M. Falcone - P. Saint-Pierre, Slow and quasislow solutions of differential inclusions, Nonl. Anal. - T.M.A. 11 (1987), 367-377. [7] C Henry, Differential equations with discontinuous right hand side for planning procedures, J. Bcon. Theory 4 (1972), 545-551. [8] F. Hiai - H.Umegald, Integrals, conditional expectations and martingales of multivalued func-tions, J. Mult. Anal. 7 (1977), 149-182. [9] E. Klein - A. Thompson, Theory of Correspondences, Wiley, New York (1984). [10] K. Kuratowski, Topology I, Academic Press, New York (1966). [II] U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. Math. 3 (1969), 510-585. [12] N .S. Papageorgiou, Convergence of Banach space valued integrable multifunction, Intern. J. Math and Math. Sci 10 (1987), 433-442. [13] N.S. Papageorgiou, Viable and periodic solutions for differential inclusions in Banach spaces, Kobe J. Math. 5 (1988), 29-42. [14] T. Zolezzi, Well posedness and stability analysis in optimization in the Proceedings of the "Fermat Days", ed. J.-B. Hiriat-Urruty, North Holland, New York (1986), 305-320. NIKOLAS S. PAPAGEORGIOU NATIONAL TECHNICAL UNIVERSITY DEPARTMENT OF MATHEMATICS ATHENS 15773, GREECE 

On the existence of $\psi$-minimal viable solutions for a class of differential inclusions

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On the existence of $\psi$ -minimal viable solutions for a class of differential inclusions

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On the existence of ψ\psiψ-minimal viable solutions for a class of differential inclusions

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On the existence of $\psi$ -minimal viable solutions for a class of differential inclusions