advertisement

Internat. J. Math. & Math. Scl. Vol. 9 No. 3 (1986) 459-469 459 EXISTENCE THEOREMS FOR DIFFERENTIAL INCLUSIONS WITH NONCONVEX RIGHT HAND SIDE NIKOLAOS S. PAPAGEORGIOU University of Illinoi: Department of Mathematics 1409 West Green Street Urbana, Illinois 61801 (Received November 21, 1985 and in revised form February 26, 1986) proe ABSTRACT. In this paper we some new existence theorems for differential inclusions with a nonconvex right -hand side, which is lower semicontinuous or continuous in the state variable, measurable in the time variable and takes volucs in a finite or infinite dimensional separable Pmch space. 1990 Mathemntics Subject Classifcation: KEY WORDS AND PI{RASFS: 34GOZ Orientor field, multfunct.on, measure of noncompactncss, measurable n,ultifunction, lower scmicontnuous multiunction, Hausdorff metric, Kamke funct ion. I. INTRODUCTION. In the recent years there }as been an increase in interest in the investigation of systems described by differential inclusions. In ay ordinary differential equation the tangent at each point is prescribed by single valued function. In a differential inclusion the tangent is prescribed by a mu]tfunction function) which is usually called a: orientor field. (set valued .’Lany problems of appl mathentics lead us to the study of d3q_nmical systems having velocities not miquc]y determined by the state of the system, but depending only loosely upon it. In these cases the classical equation f(t.x(t)) describing the dy,xmics of the system ’(t) is replaced by a relation of the form multifunction (the orientor field). called "differential inclusion". (t) F(t,x(t)) where F(.,.) is a Such a "set valued differential equation" is The initial impetus to study diffe.rential inclusions came from control theory. Then the subject found additional impo,’tmt applications in ma{hc,natical economics [I], nonsmooth dynamics [2], optimization [3], differential equations with a discontinuous forcing term [4] etc. N. S. PAPAGEORGIOU 460 The purpose of this paper is to prove existence theorems for differential inclusions governed by nonconvex valued, lower semicontinuous orientor fields which Until now, most of the existence theory for take values in a separable Banach space. differential inclusions was developed for upper semicontinuous, convex valued orientor n. fields with values in However lower semicontJnuous, nonconvex valued orienror So fields appear often in control theory in connection with the bang-bang principle. it is important to have existence theorems for differential inclusions governed by such orientor fields. 2.PRELIMINARIF. Let X (,2) be X be a separable Banach space, with We will use the following notation. a measurable space and let being its topological dual. {A _C X" Pf(X) A E For 2X\c). [A[ we set closed}. nonempty, dA(. and by sup [[x[[ xEA function from A i.e. for all A multifunction F x e X.dA(X inf aA is said to be measurable if it satisfies any of Pf(X) d we denote the distance the following equivalent conditions. is measurable for all d (x) (i) X x FC) (ii) cl{fn()}n F() (iii) (Castaing’s representation) for all X open U for all of measurable functions s.t. {fn(.)}n there exists a sequence F-(U) inverse image of U under 0 U F() E O F (U) language of measurable multifunction } e 2 (in the is called the F(’)). treatment of measurable multifuntions can be found A detailed [5] Castaing-Valadier We denote by LebesNe-Bochner S n ad Himmelberg F(’) the set of all selectors of F (n) sNce i.e. S {f(.) e F that belong to the f() e F()-a.e.}. (n) It is easy to see that this set is closed and it is nonempty if and oniy if inf Iixll xE() L+(). Assume that Y,Z are topological spaces and is lower semicontinuous V g } (1.s.c.) if and only F Y d V if for all 21a,{}. We say that Z open, {y E Y F(y) is open too. -{A-}n Finally if s-l__im n-)co By W(’) are nonempty subsets of An X (x x s-lira X, Xn,Xn we define 6 A n.n _> I}. B _C X B can be covered by finitc.ly m.nny balls of we will denote the Hausdorff measure of noncompactness i.e. if is botmded, then (B) inf{r > 0 EXISTENCE THEOREMS FOR DIFFERENTIAL INCLUSIONS r}. radius + Recall that by a Farc function we mean a function [8]). w satisfying the Caratheodory conditions (i.e. it is measurable in x),w(t,O) and continuous in u(t) of the problem 3. [7] (see This is equivalent to the Kuratowski measure of nonconq>actness also Banas-(;oebel [O,TJ’xtR+ 461 0 u(t) a.e. and such that z 0 t is the only solution O. (s,u(s))ds,u(O) F_XISTENCETHEOREIS. e The setting is the following. we consider the Lebesgue measure By X space. we ill denote T are given a fnite intervaI dr. X be lso let [O,b]. T On a separable reflexive Banach e X ith the weak topology. Cauchy problem under consideration is the following: FCt,Ct))} e Co) x xCt) By o ) we understand () for almost all t T. a solution of satisfying an absolutely continuous function X T x Our first existence result is the following" THEOII 3.1. I_f_f F TxX Pf(X) is a multifunction .t. 1) 2) for all x X, F(’,x) is measurable for all t T,F(t,-) is 1.s.c. from 3) for all x X, 4) for all B X nonempty and bounded [F(t,x)! g,(t) X into we have CFCt,B)) _(wCt,c(B)) (.) where o is the Hausdorff measure El(T) C’} a.e. with X a.e. nonco,npactness and w(’,.) is a Kamke function. () then Let PROOF: admits a solution. 1111. r mnd consider X,Br(XO) reflexivity of Br(XO) [10] x) Br..iXo Our claim is tha n- SF( x )" ’n x s t n d (fCt)) F(t,Xn) lit(t) fn(t)ll Pf(I2.(T))_X L(.) o the weM: [opology (see be the mtl] ti function is 1.s.c. tot any x n For that purpose let A straightforward application of a.e. S( C s-lm Because r}. IlX-Xoll In the sequel we will al,:ays consider we know that it suffices to sl,ow t.hat 1 Sc S(.,x). L(x) X is w-compact md metrizable for the anford-Schwartz [9], theorem 3, p. 3). Br__fXo) with the were topology’. Let L defined by {x t’(’) x SF( in x)" From Delahaye-Deel B r (Xo) Then we have fCt) Au,nknann’s selection theorem can give us for all t C T. Since F(t .) is F(t,x) f s c .) E 462 N. S. PAPAGEORGIOU Xw from O, X, into - F(t.x) C_ s-lim___ n- F(t,Xn) and so lim d n- F(t,Xn) which by the dominated convergence theorem implies that f(.) E s-lim SF(" Xn ). So we lmve. shown that SF(, f C s-lim x) n-o f(.) => n (.) 1 SF( ,Xn) which as we already said, implies the lower semicontinuity of L(-). Hence we can now apply theorem 3.1 of Fryszkowski [11] and deduce that there exists $ Br(XO) 8C x) continuous s.t. 6 LCx x e for BrCXo). set f(t.x) $Cx)C t) [,(T) and consider the following single valued Cauchy problem f(t,x(t) } x(t) (o) Let W {xC- e W x o xCt e BrCxo) CxCT for al T} and t consider the nap W defined by (x)Ct) For t,t’ E T,t _< t’ x + 0 fCs,xCs))ds. we have tbt t’ ,,x + 0 IIO(x)(t’) -b(x)(t),, 0 t f(s.x(s))ds x 0 of(S,X(s))dsII IItf(s,x(s))dsll <. ftll(s,x(s))llds < tq(s)ds :> [[Cx)(t’) -Cx)Ct)l[ < when t’ t[ ( 5, x(’) for a11 equicontinuous subset of Cx(T W. Thus v,e a deduce that (W) is an and in fact it is uniformly equicontinuous since T is a compact interval. Also we claim that W. (-) is continuous. For tha.t purpose let n (’) x Then we have" II(Xn.)(t (x)(t)ll =lix0 + _< f(S,Xn(S))ds O][fCS,XnCS)) x 0 fCs,xCs))[]ds. Applying the dominated convergence theorem we get th-t f(s.x(s))dsll x(’) in EXISTENCE THEOREMS FOR DIFFERENTIAL INCLUSIONS as . n ,Cxn) o Now consider the classical Caratheodory approximations x (t) n + 0 Note that for all n 0 > llxnCt -OCx)Ct)ll nf(s.Xn s))ds x (-) E W n iiXn Ct) CXn)Ct)’l while +C)I 463 b for and llCXn)Ct4) (Xn)Ct)l for /,1/n /,1/n Jot)fCS.XnCS}},ds Jo, CS)ds _< for l_n <0 < _< t _< t b 1/n. Thus we have that ilx as R n . Let R (XnC’)}n>_l. is uniformly equicontinuous. CRCt)) Note that given a 0 R _c (I Then since Set -* 0 0) (R) + (R) {Xn(t)}n_>1 R(t) for t E T. we deduce that Then we have __1fcs’Rcs))ds fCs,RCs))ds + ^ > CXn) IIo n n n() we can find (s)ds < s.t. a/2 for t E T,n _> n 1. Hence we have that y t_l.n CS,XnCS))ds n )_ nCa) (_ 2 sup n>n() Using this estinmte and the propoerties of [R(t)] _< Since for all Since R(O) s T,R(s) (’) t l&CS)ds y[f(s,R(s))]ds. is bounded, using hypothesis wCs.CRCs )))a.e. => ,[R(t)] w(s,(R(s)))ds. 0 and . we get that CfCs,RCs))) _< Xo,(R(O)) ( _( w(-,’) 4} we have that" is a Knmke function we must have that N.S. PAPAGEORGIOU 464 for all But recall (see [8]) that T. t TC R) So O. (R) R which means that x(’) IIxk (Xk)ll0o W. 6 IIx IIxk So is a relatively compact subset of IIx b(Xk)llo Xk(.)}k_l Cx(T ). {Xn(.)}n>_lS.t. of But we have already seen that b(x)llm. Thus finally we have that -* O. (x)ll 0 => x(t} Since the vector field of solves TCRCt)). {%(-) Therefore we can find a subsequence Xk(" sup {} x(*) f(s.x(s))ds x + 0 F{’,’}, selector of is a solves we conclude that x(’) {). Q.E.D. RENARK. When X 1) I__f F a(-),b(,) for all then () TxX Pf(X} x 6 X, F(.,x) for all a.e. with 2) F(t..} Xw is 1.s.c. from is a separable dual space ’ith into X. is finite dimensional we can have a more general boundedncss hypothesis. THEORF 3.2. PROOF" X The theorem remains true if we essume that a separable predual and 6 IF(t,:}] is measurable and a(t)llxll + b(t) EI(T T,F(t,.) t 6 is a multifunction s.t. is 1.s.c. admits a solution. [(llxoI12 + 1)e2[ilalll" + Ilblll) Let N 1] 1/2 and define the following new orientor field. [F(t,x) Mx (t II-D IF F(t.x) Then for every t T,F(t,’) and of the M-radial retraction map r(x) It is well known that measurable in t r(,) and for all - for Ilxll for llxll N > N is the composition of the multiftmction r X j []ll-Mx if f BN{O} Ilzll (_ N} F(t.’) defined by Ilxll _< 14 Ilxll is Lipschitz. x 6 X, {z X [F(t,x)[ > M So F(t,.) is 1.s.c. Clearly a(t)M + b(t)a.e. So F(.,-) it is theorem 3.1 (recall that X is finite dimensional) and so (,x(t)) T X absolutely continuous s.t. (t) by that theorem there exists x Our goal is to show that for all t e T, IIx(t)ll _< M. We proceed by a.e., x(O) x O. satisfies the hypotheses of contradiction. Suppose that there exist tl,t 2 T s.t. for t 6 (tl,t2)llx(t)ll > hi. EXISTENCE THEOREMS FOR DIFFERENTIAL INCLUSIONS ’(t) Then we bve thaz Set d--qtlx E 2 z(t) Ilx(t)ll + 1. (t)ll _< l(t)ll - FCt.[[xCt)i Then we get that d a.e. on z(t) (tl.t2) ----> [[(t)] t IIx(t)ll _< 211xCt)ll[aCt)ll -zCt) 2 + llx(t)ll > I 1 d {tl,t2) on t t (tl,t2) and get that for all _ t E on So for all we already have that x(.) {tl,t2)- IIx(t)ll (_ I, we can now integrate T z(t) z(O) / 2 [(llxol12 + l)ex’p T, llx(t)[[ _( 1. t Since (tl,t2). on (a(s) + b(s))z(s)ds. Using Gronwall’s inequality we get that for all llxCt)ll Ix(t)la.e. 2 1) + bCt)CllxCt)ll + 1)] 2[a{t) + b(t)]z(t)a.e, Since for 211x(t)ll a.e. we have that 2[aCt)Cllx(t)ll2 + "zCt + 1] aCt)hi + bCt bCt)] 211xCt)llEaCt)llx(t)ll + bCt)]a.e, Because 465 Then C2JoCaCs F(t,x{t)) t E T 1] + bCs))ds) F(t,x(t)) I/2 which implies that solves the original Cauchy problem Q.E.D. Another existence result in this direction is the following. X is finite dimensional. THEOREN 3.3. ]__f F TxX 1) for all with x 6 d Pf(X) X,F(’,x) L/ and (,) a(’) Again assume that is a multifunction s.t. is measurable and IF(t,x) a(t) @([Ixll)a.e. a positive continuous function s t + ds t 2) for all then PROOF" can find () Because bI t T,F(t,,) is l.s.c. admits a solutiou. a(,) E L+ and because of our integrability hypothesis on &(’) > ]lXol, s.t. Oa(S)ds < ’ llx 011 (s) ds. we N. S. PAPAGEORGIOU 466 Aain introduce a new orientor field F{t,x) F(.,.) defined IF(t.) [(t if Ilxll i Ilxll as before by M > M We have already seen in the proof o theorem 3.2 that F(.,-) satisfies all hypotheses of theorem 3.1. So we can find x T X absolutely continuous s.t. e F(t,x(t)) a.e. x(O) x Oar claim is that llx(t)ll M for all t e T. Suppose O. t Then we can find not. llxoll < M. [lX(to)ll > T s.t. o t* e T Hence we can find s.t. M. On the other txnd we have M for all llx(t)ll llxoll llx(O)ll [O,t’]. So t we can write (t} F(t,x(t)}a.e. on => [[(t}l[ < a(t}#([[x(t}[I}a.e, => -Itlx(t)ll on _< a(t)#(llx(t)ll)a.e, => dlx(t )[ _( 2 [O,t’] E0. t’] on (s)ds (s)ds J,x0,(s IIx(t)ll g M for all which implies tlt x(’) solves (). a contraciction to our choice of F(t,x(t)) F(t,x(t)), M. Thus t 6 T and so Q.E.D. REMARKS" T 1) [0,1] [1,2] If R+ T then we divide it into subintervuls we consider the Cauchy problem we consider gain () but () and find a solution with initial condition this way we obtain a sequence of partial solutions x(1) {Xn(’)}n which vhen pieced together give us the global solution on 2) I the domain o F(’,’) is TXBr(XO), T [n-l,n]. Xl(- ). Xl(1 ). (h, Then on T2 Continuing defined on Tn n )_ +. then local versions o those results are valid. 3) The above theorems as well as the one tl’t follows extend significantly earlier results obtained by Bressan e [12], Kacz3mski-Olech [13] and Lojascwicz [14]. will conclude this work with an existence result concerning continuous orientor-fields on separable Ba_ch space concept of a semi-inner product. X. 1, But first we need to introduce the EXISTENCE THEOREMS FOR DIFFERENTIAL INCLUSIONS 467 x X be a Banch spce and X its dual. Consider the map J X 2 defincd by J(x) Thanks to the Hahn-ch { X (x.x) ,x, 2 X. Using J(-} we can define the semi-lnner product theorem J(x) for all x by (...)_ XxXLet ,xa,2}. Inf{CY.X} y Cx.Y}_ J(y)}. For more details about semi-inner products the reader can consult Dimling [15] (p. 33). Now let T [O.b] and let X be any separable Banach space. By h(’..) we Pf(X). will denote the Hausdorff metric on I_[ F THEOREM 3.4. TxX Pf(X) is a multifunction s.t. 1) for all x X, F(..x) is measurable 2) for all x,y E X, h(F(t,x),F(t,y}) < k(t)llx- yl[ 3) SF(..Xo 4) for all then () y e F(t,x) (y,x)_ < cCt)EIIxll2+l] M2 Let F(’,’) defined by [.[]Xo]12+l]e 1 e LI+cT)- Again we introduce the new orientor field 1. f J](t’x) F(t.x) Ct.,-) Clearly FCt.x FC’,x) is measurable. Also FCt.r(x)). So for any x,y _ h((t,x).(t,y)) Furthermore note that S g llXol if Ilxll if Ilxll r(’) if > M is the M-radial retraction map, X we have" 6 r(y)ll < M and so 2k(t)lix-y[[. F(t,Xo) which is equivalent to saying that F(-.Xo) F(t,Xo}. tnfllyll ze(t,Xo So we can apply theorem LI(T) "t" (t.x(t)) x(O} Let ltence by hypothesis of Muhsinov [16] and get that the Cauchy problem (t) admits a solution M h(F(t.r(x}),F(t,r(y))) k(t) lir(x) 3) coo L+*(T) admits a solution. PROOF" then with k(-) e with x(*) be such the original Cauchy problem. x o a solution. I"o show that let u(t} We will show that 2 [Ix(t)ll + 1]. x(-) solves Then we have" N. S. PAPAGEORGIOU 468 (().x())_ <_ c()[x() => u(t) _< u(O) c()u() z] + c(s}u(s)ds. + Applying Gronwa11’s inequality we get that Ilcll u(t) <_ e 0112 [llx --> lu(o Ilcll + 1]e x()2 < 2 => x(t) _< But then from the definition of FCt,x(t)) F (t0xCt)) => x(’) tET we bve that for all F(.,’) solves (). Q.E.D. In this paper we extended the works o Kaczyski-Olech In particular theorems 3.1 Lojasicwicz [14]. [13], Brcssan [12] and and 3.4 provided infinite dimensional versions of those results,which are important in studying distributed parameter On the other control systems, characteristic of mechanics and mathematical physics. hand in theorems 3.2 -und 3.3 which are finite dimensional, we have less restrictive hypotheses than [12], [13] and [14]. Specifically our orientor field satisfies Caratheodory type conditions while in [12] and [14] F(*,*) lower-semicontinuous and in [13] Furthermore our boundedness hypothesis is more general than thie of o radius H > O, [12] mid x. [14] [13] while in is integrably bounded. ACKNOWLFADCHENT: suggestions. I would like to thank the referee fcr his constructive o the University of"Pavia-Italy. and N.S.F. Grnt D.H.S. 8q03135. J.P. criticism and This work was done while the author was visiting the Hathematics Department 1. 2. is jointly iz is Hausdorff continuous in zhe state variable where the orientor field stays within a fixed ball F(*,*) F(’,,) Financial support was provided by C.N.R. Aubin-A. Cellina: "Differential Inclusions" Springer, Berlin (19S.i). "Optimization and Hoasmoorh Anal;dts" Wiley, New York (19S3). F.H. Clarke: EXISTENCE THEOREMS FOR DIFFERENTIAL INCLUSIONS 3. L. Cesari" "Optimization-Theory and Applications" 469 Spriner, New York (19S3). 4. A. Filippov" "Differential equations with discontinuous right hand side" Translations of the A.M.S. 42(1964) pp. 41-46. 5. C. Castaing-M. Valadier" "Convex Analysis and Measurable Itifunctions" Lecture Notes in Math, Vol. BSO, Springer, Berlin (1977). Fund. Math $7(1975) pp. 52-71. 6. C. Himmelberg" "Measurable relations" 7. K. Kuratowski" Sur les espaces comp]etes" 8. 9. Fund. lath 15(1930) pp. 301-309. J. Banas-K. Goebe" "Measures of noncompactness in Banach spaces" Dekker Inc., New York (1980). N. Dunford-J. Schwartz" "Linear Operators I" Wiley, New York Marcel (1958). 10. J. P. Delabaye-J. Denel" "The continuities of the point to set maps, definitions and equivalences" th. Progr. Study 10(1979) pp. 8-12. 11. A. Fryszkowski" "Continuous selections for a class of nonconvex multivalued maps" Studia Fth IO(XVI(19B3) pp. 163-74. A. Bressan: "On differential relations with lower continuous right hand An existence theorem" J. Diff. Eq. _7(1980) pp. 89-97. 13. H. tczynski-C. Olech" "Existence of solutions of orientor fields with 12. nonconvex right hand side" 14. Annales Pol. th. 29__(1974) side. pp. 61-66. S. Loasiewicz" "The existence of solutions for lower semicontinuous orientor Bul. Acad. Pol. Sc. 2__.8(1980) pp. B3-87. ields" I5. K. Deimling" "rdinary Differential Equations in tkmach Spaces" Springer Berlin’(19?).’ Lecture Notes in Math, Vol. 596, 16. A. Muhsinov" 15,_(1974) "On differential inclusions in Banach spaces" pp. 1122-25. Soy. Math. Dokl.