Evolution equations governed by the sweeping process

Set- Valued Analysis 1: 109-139, 1993 @ 1993 KIuwer Academic Publishers Printed in the Netherlands 109 Evolution Equations Governed by the Sweeping Process C C A S T A I N G 1, T R ) O N G XU/~N DI)C H A 2, and M V A L A D I E R Ddpartement de Math~matiques, case 051, Universitd Montpellier II, F-34095 Montpellier Cedex 5, France Department of Mathematics, University of Hanoi, Hanoi, Vietnam (Received: November 1992) Abstract This paper is concerned with variants of the sweeping process introduced by J.J Moreau in 1971 In Section 4, perturbations of the sweeping process are studied The equation has the form X'(t) E -No(t)(X(t)) + F(t, X(t)) The dimension is finite and F is a bounded closed convex valued multifunction When C(t) is the complementary of a convex set, F is globally measurable and F(t, ) is upper semicontinuous, existence is proved (Th 4.1) The Lipschitz constants of the solutions receive particular attention This point is also examined for the perturbed version of the classical convex sweeping process in Th 4.1 t In Sections and 6, a second-order sweeping process is considered: X" (t) E -Nc(x(t)) (X' (t)) Here C is a bounded Lipschitzean closed convex valued multifunction defined on an open subset of a Hilbert space Existence is proved when C is dissipative (Th 5.1) or when all C(z) are contained in a compact set K (Th 5.2) In Section 6, the second-order sweeping process is solved in finite dimension when C is continuous, Mathematics Subject Classifications (1991) Primary: 35K22 Secondary: 34A60, 34G20 Key words: Evolution problems, differential inclusions, sweeping process, perturbations, Lipschitz solutions, second-order sweeping process Introduction The sweeping process (in French, rare) was introduced by J.J Moreau in 1971 ([29,30] and also [31-37, 5-12, 42 46]) It is a kinematical evolution problem closely related to the modelization of elasto-plastic materials For this application, infinite-dimensional Hilbert spaces are needed But the intuitive characteristics of the sweeping process are easier to apprehend in finite-dimensional Euclidean space Let C(~) be a m o v i n g closed set and a E C(0) A trajectory is a function t ~-+ X ( t ) verifying X ( ) = a, Vt, X ( t ) E C(t) and X ' ( t ) is an inward normal vector to C(~) at X ( t ) (this implies X'(~,) = when X ( t ) is interior to C(~)) W h e n C ( t ) is convex, uniqueness holds Convexity was the framework o f J.J M o r e a u (note that usually the evolution o f an elasto-plastic material is not unique Even for a bar, the point(s) where the elastic limit is first attained may be several [35] There is no contradiction there: that only proves that the evolution of an elasto-plastic material is not exactly a sweeping process) In [43-45], the third author studied some cases o f the sweeping process without convexity (in finite dimension) In 110 C CASTAINGET AL these sweeping processes, the most comfortable hypothesis is the Lipschitzity of C with respect to the Hausdorff metric When C is absolutely continuous or 'of bounded variation and continuous' a 'change of watch' (See [32] and [38] For an example of this technique, see Proposition 4.3 in Section 4.) can in some sense transform the problem into a Lipschitz one But the case where C is only continuous is different: it needs the hypothesis int C(t) ~ ~ (and convexity) The sweeping process under these hypotheses was solved, after the pioneer works of H Tanaka [40] and C Castaing [9,10], by M.D.P Monteiro Marques [25-27] In Section 3, several useful results are recalled In Section 4, perturbations of the sweeping process are studied Many predecessors (doing their research in Montpellier) have to be referred to (A Gamal [18-20], S Bahi [3], M.D.P Monteiro Marques [24], N Larhrissi [22], Duc Ha [17]) The equation has the form X'(t) ~ - N c ( t ) ( X ( t ) ) + F(t, X ( t ) ) The dimension is finite and F is a bounded closed convex-valued multifunction In the nonconvex case (more precisely, C(t) is the complementary of a convex set), existence is proved (Theorem 4.1) First, global upper semicontinuity of F is assumed, then this is weakened into global measurability of F and upper semicontinuity o f F ( t , - ) The Lipschitz constants of the solutions receive particular attention This point is also examined for the perturbed version of the convex sweeping process in Theorem 4.1~ In Sections and 6, a second-order sweeping process is considered: x"(t) -Xc(x(,i)(x'(t)) This was initiated by the first author in [12] (for an exposition, see [28, Ch 5]) Similar equations appear in the modelization of dry friction [37, (5.6) and (8.3)] Here, C is a bounded Lipschitzean closed convex-valued multifunction defined on an open subset of the Hilbert space Existence is proved when C is dissipative (Theorem 5.1) or when all C(x) are contained in a compact set K (Theorem 5.2) In Section 6, the second-order sweeping process is solved in finite dimension when C is continuous Then Y : X ~ is a BV function and the foregoing equation has to be replaced by dDY IDY] - a.e dlDY -~[(t) C -Nc(x(t))(Y(t)) , where D Y is the Stieltjes measure or differential measure of Y, [DY[ its variation measure and the fraction denotes the Radon-Nikodym derivative The proof is new and relies on an approximation of C by a decreasing sequence of Lipschitzean multifunctions Notations and Preliminaries We will use the following notions and notations We denote by EVOLUTION EQUATIONS GOVERNED BY THE SWEEPING PROCESS 11] H a separable Hilbert space, (x, g) the scalar product of z and y, r) the closed ball with center z and radius r, B(x, r) the open ball, cc(H) the set of nonempty closed convex subsets of H, ck(H) the set of nonempty convex compact subsets of H, cwk(H) the set of nonempty convex weakly compact subsets of H 5( IA) the indicator function of a subset A of H (it takes value on A, + e o elsewhere), ~*(.tA) the support function of A, - projA(x ) the projection o f x on A E cc(H) One has [1, p 23]: - - - / ) ( x , - - y=projz(x )e=~yEA - and V a E A , (x-y,y-a} NA(y) the normal cone to A E cc(H) at y (as usual in convex analysis, one considers outward normal vectors) If y ¢ A, NA(y) = One has [1, p 220] or [2, p 168]: n E NA(y) ¢¢, [y E A and (n,y) = ~5*(nlA)], y = projA(x ) ¢* x y E NA(y) • - - - >0 (2.1) If A is a nonempty closed subset of R d and x E R d, projA(x ) denotes the set of points of A which are proximal of x Then, for any y belonging to proja(x ), x - y belongs to the normal cone to A at y in the sense of Clarke [15] This normal cone is still denoted by NA(y) (for an extension of this to infinite-dimensional Hilbert space, see [23,4]) int A the interior of A If A and B are subsets of H, the excess of A over B is e(A,B) = sup{(d(a,B) : a E A} and their Hausdorffdistance is h(d, B) = max(e(d, B), e(B, d)) - - - The excess e(A, {0})is denoted by [A[ (then Id[ = sup{1]a[[ : a E d}) R d is endowed with its canonical Euclidean structure The scalar product of z and z I is denoted by (z, zl/ If I denotes the interval [0, T] of R, )~ is the Lebesgue measure on I and T~ (I) is the o algebra of all Lebesgue measurable subsets of I A subdivision of [t3,T] (T > 0) is a finite sequence ( t o , , tn) such that -t0 such that Vh E]0, r]], X(t + h) ~ K(t + h) Indeed, if such an ~ did exist, for any h El0, 7], X(t + h) would be interior to C(t + h), hence Y(t + h) = So Y(t) would be zero Then, as in [43, Th 6, pp 8.16-17], there exists a sequence (h~)n converging to in ]0, T - t] such that X(t + hn) E K ( t + hn) There exists e • ]0, T - t] ~ R d converging to in + such that X(t + h) = X(t) + hX'(t) + he(h) Now we have e(K(t + hn), K(t)) > d(X(t + hn), If(t)) > d(X(t) + hnX'(t), K(t)) - hn IIc(hn)lt • Suppose first (X'(t), Y(t)) _> Thanks to the inclusion K(t) C {~ " (Y(t), ~ X(t)) >O From Vn, e(K(t + hn),tq2(t)) ~ klhn, follows IIY(t)ll + IIZ(t)ll cos ~ < k~ Since IlX'(t)ll a = IllY(t)ll + Ilz(t)]] cos a[ + (lIZ(t)ll sin a) , one has ][x'(t)]] < k~ + k~ Finally, if (X'(t), Y(t)} < 0, that is, if llY(t)ll + llz(t)tt cos ~ is < o, one has lllY(t)li + llZ(t)lt cos ~1 -< lliZ(t)H hence again lix'(t)ll _< k + k2 H " 7-(O /~K(O X'(t) x x Case (Y(t), Xt(t)) < O cos c~l 118 c CASTAINGET AL (2) N o w we turn to the proof of the last assertion First step Assume first that (iii) For the existence we take T of [0, 1] • S ~ = ( t ~ , , t~,~) X~(t~) = a, z~ be a point in F is globally upper semicontinuous and satisfies = Let (S n)n> be the sequence of subdivisions with t~z = i2 -n Let n be a positive integer Let F(trd, X~(trd) ) and y~ E projc(ff)(a + - n z ~ ) , where projc(t?)(a + - n z ~ ) denotes the set of points of C(t] ~) which are proximal of a + 2-nz~L For t E [t~, t]~] put t - t~ t? - t t I - to t? - t~ Then X~n(t) _ y _~_ a = 2n(y ~ _ a) and X ' ( t ) - z~ = 2**(y~z - (a + - n z ~ ) ) E - N c ( f f ) ( X ( t ~ ) ) Let w~ E projc(ff)(a) Since C is kl-Lipschitzean ([43, L e m m a 4, p 8.15]) and t~ - t~ = -~, one has IIw~ - all k12 -~ One has Ily? - all ~ Ily~ - a - 2-nz~[[ + 2-~llzg~ll < [[wg - a l l + 22-~11zgit , Consequently, since l[z~tl _ k2, ] l y r - all < (k, + 2k2)2 - n , hence llX'(t)lt _< (kl + 2k2), Then, by induction on i (0 < i < 2" - 1), there are two finite sequences (z~)0_ is a Cauchy sequence for the metric of uniform convergence By extracting subsequences, we can suppose that EVOLUTION EQUATIONS GOVERNED BY THE SWEEPING PROCESS 129 and ( ~)r,>] converge for the cr(L 1, L ~ ) topology to X ~and Y~, respectively Then almost everywhere X'(t) E / ) (0, 1) and Y'(t) E / ) (0, 1) Put t [o,1], x(t)=xo+f x'(s)ds, t VtE[O,I], Y(t)=yo+/ Y'(s)ds Then lim,~ ,~ Xn(t) = X(t) uniformly on [0, 1] Since X~(t) = Yn(5~(t)) and l i m n ~ Y~(5~(t)) = Y(t) for the or(H, H) topology, by Lebesgue's theorem, we get t VtE[0,1], t xo+ / X~(s)ds=xo+ f Yn(6n(s))ds 0 +xo + ,/ Y(s) ds for the cr(H, H) topology It follows that t VtE[O, 1], X(t)=xo+ f Y(s)ds Since y.(o.(t)) e c(xn(o.(t))) and lim ~ -~ 0 Xn(On(t)) = X(t) for the norm topology, we get Y(t) E C(X(t)) for all t E [0, 1] because the graph of C is closed in f2 x He To finish the prooL we check that Y~(t) ~ -No(x(t))(Y(t)) a.e Let V(x,y) E f2 x H, ~(x,y) = 6*(ylC(x)) Then ~ is lower semicontinuous on f2 x Ho- and ¢(x,-) is convex on H Moreover, for all (x, y) E f2 x / ) ( , 1), ]~(x, Y)I < Applying Theorem of [47], we obtain lirainfn +oof 5*(-Y~(t)IC(Xn(t)))dt > f 5*(-Y'(t)lC(X(t)))dt'(5"l'lO) 130 c CASTAINGETAL Moreover by [14, pp 221-222], we have lim inf S ?'t, + o O (Ytn(t)' Yn(t))dt > f (Y'(t), Y(t))dt (5.1.11) We repeat here the main argument of [14, pp 221-222]: f (Y'(t), Y~(t))dt t = j ,,-ilY'(,~, o(,°(,~)>.,i 0 }~,_ with xo + TB(O, 1) C t2 Assume that there is a convex compact subset K of/3(0, 1) such thatVx E t2, C(x) C K Then there are two 1-Lipschitzean mappings X : [0, T] -+ f~ and Y : [0, T] -+ H such that t Vt E [0, T], X ( t ) = xo + f Y(s) ds, (5.2.1) w e [0,T], Y(t) = vo + f V'(s)ds, (5.2.2) t Vt E [0, T], Z(t) E C ( X ( t ) ) , (5.2.3) Lebesgue a.e Y'(t) E -Nc(x(t))(Y(t)) (5.2.4) Proof An inspection of the proof of Theorem 5.1 shows that it is enough to obtain the relative compactness of the sequence (Xn)~_>l of approximate solutions for the topology of uniform convergence Since Vx E f~, C(x) is included in the convex compact set K , for any t, Xn(t) belongs to xo + T K and (-X~)r~>l is relatively compact thanks to Ascoli's theorem Therefore, by extracting a subsequence, we 132 c CASTAINGET AL l can suppose that ( X n)n_>l converges to X ~for cr(L 1, L ~ ) and (Xn)~>_l converges uniformly to X with t vt, [o, 11, x(t)=xo+f so that we can repeat the arguments of Theorem 5.1 to finish the proof Lipschitz Approximation for the Second-Order Sweeping Process Existence for the second-order sweeping process when C is only assumed to be continuous is due to [12] (for another proof see [28]) The proof here is new Lipschitz approximations of multifunctions were obtained by Valadier [41] and A Gavioli [21] THEOREM 6.1 Let f2 be a nonempty open subset of R d, (Cn)n>_l a decreasing sequence ofLipschitzean multifunctionsfrom f~ to ck(R a) and C : f2 -+ ck(R d) a continuous multifunction satisfying: (i) Vx E f2, lira h(C~(x), C(x)) = O, n ~oo (ii) there exists ro > such that Vx Efl, C(x) D B(O, to), (iii) sup ICl(x)l < xEf~ Let xo E f~, Yo E C(xo) and T > with xo + TJ~(O, 1) C fL For each n > 1, let (Xn, Yr~) be a pair of Lipschitzean mappings solving the second-order sweeping process associated to the multifunction Cn, that is t V t E [O,T], Xn(t) = xo q- / Yn(s) ds , (6.1.1) t Vt E [0, T], Y~(t) = Yo + / Y~(s)ds, (6.1.2) Vt E [0, T], Yn(t) E Cn(X~(t)), (6.1.3) Lebesgue a.e Y~(Â:) E -Nc~(x,(~))(Yn(t)) ã (6.1.4) Then there exist a subsequence (Xn~,Ynk)k>l, a Lipschitzean mapping X • [0, T] -+ R d and a B V and continuous mapl~ng Y " [0, T] + R d such that Xn~ converges uniformly to X and ~.k converges uniformly to Y and (X, Y) is 133 EVOLUTION EQUATIONS GOVERNED BY THE SWEEPING PROCESS a solution of the second-order sweeping process associated to the multifunction C in the following sense: t vt < [o, T], X(t) = xo + f V(s) ds , v(o) = v , vt ~ [O,T], Y(,) e C(X(+)), dDY ]DYI - a.e dJDY ~I (t) ~ -Nc(x(t))(Y(t)) Remark In the theorem, C is assumed to be continuous By a recent result due to Gavioli [21], as soon as C is upper semicontinuous with sup IC(x)l < c , xE~2 there exists a decreasing sequence (C,+)~1 of Lipschitzean multifunctions from f2 to c k ( R d) satisfying lira h(Cn(x), C(x)) = Vx C f2, and sup ICl(x)l < o c n +c¢ xE~2 Proof The existence of (Xn, Yn) follows from (iii) and Theorem 5.2 By the proof of that theorem, the sequence (Xn)n> is equi-Lipschitzean Therefore, there exists a subsequence (X~k)k>l, still denoted by (A~)n>_l in this proof, which converges uniformly to a 1-Lipschitzean mapping X : [0, T] -+ R d By (ii) and Theorem 3.2, T P sup [ tt~(t)ild+ < ~ n_>l J0 Now we prove first that (Yn)nkl (more precisely (Ynk)k>_l) is a Cauchy sequence for the norm of uniform convergence on [0, T] Note that for any t ~ [0, T] and any positive integers m and n, we have t IIYn(t) - }%(t)[[ (Y,,(s) - Ym(s), o For m _ m > m0 implies W e [O,T], h(C~(X,~(s)), G(X,&))) l -I sup h(C.,(Xm(~)), C~(X,,(~))) sE[0,T] (6.t.10) 136 c CASTAING ET AL Finally, by combining (6.1.6), (6.1.9) and (6 I 10), we have Vm, n such that n > m >_ m0, T IIY n ( ~ ) Y m ( ~ ) l l < e (sup IIY~'(t)il "n> - dr) Hence, (Y;~)~>1 is a Cauchy sequence Consequently, (Yn)~> converges uniformly to a continuous function of bounded variation Y with T T f d]VY[ < sup / n>l - HY'(QHdt Since Y~(t) E Cn(Xn(t)) for all n > and all t E [0, T] and since h(C~(x), C(x)) = O, we have Vt E [0, T], Y(~) E C(X(t)) It remains to prove the inclusion dDY IDY1- a.e dIDY ~(t) E -Nc(x(t))(Y(t)) lim (6.1.11) Thanks to the characterization (2.1) of normal vectors, (6.1.11) is equivalent to forO