Set-valued Solutions to the Cauchy Problem for Hyperbolic Systems of Partial Differential Inclusions

We prove the existence of global set-valued solutions to the Cauchy problem for partial differential equations and inclusions, with either single-valued or set-valued initial conditions. The method is based on the equivalence between this problem and problem of finding viability tubes of the associated characteristic system of ordinary differential equations or differential inclusions. On de'montre l'existence de solutions multivoques globales du problime de Cauchy pour les systt?mes hyperboliques du premier ordre d'e'quations ou d'inclusions aux de'rive'es partielles, pour des conditions initiales univoques ou multivoques. La me'thode est base'e sur l'e'quivalence entre ce problime el celui de l'existence de tubes de viabilite' pour le systime caracte'ristique d'e'quations diffe'rentielles ordinaires ou d'inclusions diffe'rentielles. Set-Valued Solutions to the Cauchy Problem for Hyperbolic Systems of Partial Differential Inclusions Jean-Pierre Aubin & H616ne Frankowska


FOREWORD
We prove the existence of global set-valued solutions to the Cauchy problem for partial differential equations and inclusions, with either single-valued or set-valued initial conditions.
The method is based on the equivalence between this problem and problem of finding viability tubes of the associated characteristic system of ordinary differential equations or differential inclusions.

Introduction
It is well known from the method of characteristics that first-order systems of hyperbolic partial differential equations may and do possess set-valued solutions, the set-valued character of a given solution providing an explanation for shocks.
One can use the differential calculus of set-valued maps for looking for global set-valued solutions to such hyperbolic systems of both partial differential equations and inclusions.
We shall prove the existence of a largest set-valued solutions with closed graph, which is unique (among closed graph single or set-valued solutions) whenever the characteristic system enjoys the uniqueness property.
The method we use is based on the equivalence between solutions u(t , x) = (ul(t, x), . . . , u,(t, x)) to the system of partial differential equations n v j = 1, ..., m, 0 = auj(t7x) +C d"j(t,x) fj(t,x,u(t,x)) -gj(t,x,u(t,x)) dt i=l dx; and bilateral viable tubes1 P(.) under the characteristic system (1) The link between (single-valued or set-valued) solutions to (1) and tubes bilaterally viable under the characteristic system (2) is given by the relation

'We recall that a solution t -(x(t), y(t)) E X x Y to (2) defined on [0, +oo[is viable in the tube P if v t 2 0, (~(t), ~(t)) E P(t)
A tube P is bilaterally viable under the system (2) if, for all to 2 0 and (xtD, yt,) E P(to), there exists at least one solution (x(.), y(.)) to the differential system (2) satisfying (x, y)(to) = (xt,, yt,) which is viable in the tube P. Therefore, the existence of solution to the Cauchy problem for (1) satisfying the initial condition v x E X, u(0, x) = uo(x) is equivalent to the existence of a tube bilaterally viable under the characteristic system (2) satisfying the initial condition Our objectives are twofold: to prove the equivalence between Cauchy problems for hyperbolic systems of partial differential equations and initial value problems for viable tubes of ordinary differential equations on one hand, to prove the existence of the largest tube bilaterally viable on the other hand and to characterize it.
This equivalence allows also to transfer other properties of viable tubes to corresponding properties of solutions to partial differential systems.
There are obvious advantages in doing so. First, dealing with graphs of solutions, we do not have to worry about the univocity issue: the viable tube provides the graph of a solution, single-valued or set-valued. We can tackle for instance the question of the existence of a largest solution as well as the existence of minimal solutions containing a given function.
The other advantage is that we can treat in the same way not only systems of partid differential equations, but also partial differential inclusions, since the results about viable tubes are still valid for ordinary differential inclusions First-order systems of partial differential inclusions arise naturally in control theory (see [7,9,8]).
For instance, we shall prove a stability theorem: the graphical upper limit2 of a sequence of solutions U, is still a solution and that in the time independent case, the graphical upper limit of the solutions U(t, -) when t + ca is a solution to the stationary problem.
We shall provide an explicit formula in the decomposable (set-valued) case from which we derive useful estimates. They are applied later on to prove the existence of single-valued Lipschitz contingent solution to the Cauchy problem for systems of partial differential inclusions on a small time interval by using fixed point arguments.
'The graph of the graphical upper limit UI := ~irn?,,,~, of a sequence of set-valued maps U,, : X -u Y is by definition the graph of the upper limit of the graphs of the maps un.

Cauchy Problem for Viability Tubes
The differential calculus for single-valued maps, including inverse function theorems, can be extended to set-valued maps.
We recall that the contingent derivative DU(x, y) of a set-valued map Chapters 4, 51 for more details on contingent cones and differential calculus of set-valued maps.
We say that a set-valued map P : t E [0, +oo[-.., P(t) c X is a tube, and that a tube is closed if its graph is closed.
We shall say that a set-valued map F is a Marchaud map if it is nontrivial, upper semicontinuous, has compact convex images and linear growth.
In finite dimensional spaces, this amounts to saying that Let us consider a sequence of set-valued maps F, : X -+ Y. The setvalued map Ffl := ~im~,,,~, from X to Y defined by is called the (graphical) upper limit of the set-valued maps F,.
We derive the following characterization of bilateral viability:  Proof -Let P(-) be a backward viability tube and xo belong to P(to). First, consider the tube pt0(s) := P(tos) defined by We observe that because one can check easily that if and only if Second, we consider the set-valued map Gt0 defined by It is a Marchaud map since F is assumed to be a Marchaud map. Then, we observe that P is a backward viability tube if and only if the graph of pt0 is a viability domain of Gt0 . Therefore, Theorem 3.3.5 of VIABILITY THEORY, [2,Aubin] implies that this is equivalent to say that the graph of pt0 is viable under G~,.
This means that for every to E [0, +m[, xo E P(to), there exists a solution z(-) to the backward differential inclusion zl(t) E -F(tot, z(t)) starting at xo at time 0 and viable in the tube t --, P(tot) for all t E [O,to]. By setting x(t) := z(tot) when t E [O,to], we infer that x(.) is a solution to the differential inclusion x' E F(t,x) starting at x(0) = z(to) E P(0) and satisfying x(to) = xo.
We show next that the upper graphical limit Pfl of a sequence of tubes Pn bilaterally viable under F is still bilaterally viable under F. Let x belong to P"(t). This means that t is the limit of a subsequence t,~ and that x is the limit of a subsequence X,I E Pnl(tnt). Since the tubes Pn are bilaterally viable under F, there exist solutions ynl(.) to differential inclusion (4) starting at Pnl(0), satisfying ynl(tnl) = xnl and viable in P,I. Theorem 3.5.2 of VIABILITY THEORY, 12, Aubin] implies that these solutions remain in a compact subset of C(0, +m; X). Hence a subsequence (again denoted) ynl(-) converges uniformly on compact intervals to a solution y(.) to differential inclusion (4) starting at Pfl(0) and satisfying x(t) = x. Since ynl(t) belongs to Pn,(t) for all n', we deduce that y(t) does belong to Ptl(t) for all t 2 0.
When the sequence Pn is decreasing, we know that its upper limit is equal to the intersection of the Pn : Pfl(t) = n Pn(t). n>O Therefore, by Zorn's Lemma for the inclusion order on the family of closed tubes bilaterally viable under F and satisfying Q(0) = P(O), we deduce that any closed tube Q starting at P(0) is contained in a minimal closed tube bilaterally viable and starting at P(0).
For Lipschitz maps, we recall a characterization of the invariant tubes. Theorem 11.6.2 of VIABILITY THEORY, [2,Aubin] Equality holds true if the set-valued maps F(t,.) are A-Lipschitz for every Proof -The reachable tube RK (-) is obviously invariant and backward viable under F: Indeed, if xo E RK(tO), there exists by definition a solution x(-) to the differential inclusion (4) starting from K at time 0 and passing through xo at to. Furthermore, every solution y(-) to differential inclusion (4) starting at xo at time to, concatenated to x(.) restricted to the interval [0, to] being a solution to our differential inclusion starting at K, RK(-) is invariant.
Let us consider a closed tube P c RK invariant under F starting at K.
We claim that it is equal to the reachable tube. Otherwise, there would exist x, E RK(s) such that x, 4 P(s). Since the reachable tube is backward viable, there exists a solution x(-) to the differential inclusion (4) starting from x(0) E K such that x(s) = x,. But starting from x(O), the solution is viable in the tube P since it is invariant under F and satisfies P(0) = K. Therefore x(s) belongs to P(s), a contradiction.
Let now P be any closed bilateral viability tube starting from K at time 0 and let us check that it is contained in the reachable tube. For that purpose, take xo E P(to). By Proposition 2.2, we know that there exists at least one viable solution starting at P(0) = K and passing through xo at time to. Hence xo E RK(to). By Proposition 2.2, we know that the graphical upper limit is a bilateral viability tube. Proposition 7.1.4 of SET-VALUED ANALYSIS, 15, Aubin & Frankowska] implies that we infer that it is a bilateral tube starting at KH, and thus, contained in RKr.
Conversely, let us choose xt, E RKt(to). Then there exist a solution x(.) to (4) starting from some x(0) E KH and satisfying x(to) = xt, and a subsequence (again denoted by) xn E Kn converging to x(0). By the Filippov Theorem4, there exist solutions x, (-) to (4) starting at x, such that We .thus derive from Gronwall's Lemma that xn(to) E RK,(to) converges to Proof -Indeed, we know that every solution x(.) to differential inclusion (4) satisfies v t 2 to, IIxl(t>ll 5 c(IIx(t0)ll + 1) ec (t-to) so that Therefore, every solution x(-) to differential inclusion (4) starting from the closed subset QK(to) at time to satisfies Since QK (to) := K + I: 1 1 K 11 + l)(ecto -1) B, we infer from these two inclusions that x(t) remains in the tube QK(t) for t > to.

Contingent Solutions
Consider two finite dimensional vector-spaces X and Y and two set-valued maps F : [0, +oo[xX x Y -+ X, G : [0, +oo[xX x Y -+ Y. Let DU(t, x, y) denote the contingent derivative of U at a point (t, x, y) of the graph of U.

Definition 3.1 We shall say that a closed set-valued map
is a forward (contingent) set-valued solution to the partial differential inclusion (6).
It is said to be a backward (contingent) set-valued solution to (6)  Naturally, whenever the contingent derivatives DU(t , x, y ) are even, then forward and a backward solutions do coincide. When U = u : [0, +oo [x X I-+ Y is a single-valued map with closed graph, the partial contingent differential inclusion (6) becomes Let the initial condition Uo : X -+ Y, a single-or set-valued map be given.   > 0, for every pair xi, yi, ui E F(t,xj, yi) and vi E G(t,xi,yi) (i = 1, 2), we have Furthermore, we can associate with any selection V(t, x) C U,(t, x) satisfying V(0,x) = Uo(x) a minimal solution P c U, to (6) satisfying the same initial condition and containing V.
Proof of Theorem 3.2 -By Theorem 2.4, the reachable tube RK(*) : R+ -A X x Y starting at K := Graph(Uo) at time t = 0 for the system of differential inclusions (3) is the largest closed bilateral viability tube of the system of differential inclusion (3). The map Urn(., .) : R+ x X -A Y defined by the method of characteristics is equal to Then U,(O, -) = Uo(.) and Graph(U,) = Graph(RK). Since RK(-) is a viability tube, its graph is a viability domain of the set-valued map (1) x F(t, x, y) x G(t, x, y). This amounts to saying that Since TGra ~h(ua )( t, x, y ) = Graph(DU, (t, x, y)), the above relation means that In the same way, to say that RK is a backward viability tube amounts to saying that Let us consider any closed selection V of the solution U, to the Cauchy problem for (6) with which we associate a closed tube Q defined by Q(t) := Graph(V(t, .)). Then there exists a minimal bilateral viability tube containing the closed tube Q, with which we associate a minimal set-valued solution to (6) containing this selection V.
Let us derive the corollaries in the case of hyperbolic systems of partial differential equations. Dealing with set-valued initial conditions is justified for instance to study the case when disturbances Uon(x) = uo(x) + B of the initial condition uo(x) are involved. This approximation procedure makes sense since we obtain the following stability result with respect to the initial conditions: If we assume furthermore that the set-valued maps F(t, ., -) and G(t, a, .) are A-Lipschitz, then Ub(t, -) = U,(t, .).
The proof follows from the second statement of Theorem 2.4.
We also derive the following asymptotic result: ii) Y1(t> E G(x(t), Y(t)) i.e., a solution Urn(.) to the stationary problem.
We also deduce the following characterization of the solution U, : Theorem 3. 6 Let us assume that the maps F(t, -, a) : X x Y -X and G(t,.,.) : X x Y Y are A-Lipschitz maps with compact values. Then, for any initial condition Uo : X -Y, (t,x) -U,(t,x) is also the solution to satisfying the initial condition It is minimal in the sense that any closed set-valued map U contained in U, , satisfying (10) and the same initial condition is equal to U,.
Proof -We know that the reachable tube RGraph(u (-) is an invari-

0)
ance tube thanks to Proposition 2.3, the smallest of the invariance tubes starting at Graph(Uo). We have defined U, as the set-valued map the graph of which is equal to RGraph(uo) (.). By Invariance Theorem 11.6.2 of VIA-BILITY THEORY, [2,Aubin], this graph is an invariance tube. This means that Since TGraph(ua)(t, x, y) = Graph(DU,(t, x, y)), this is equivalent to say that U, satisfies property (10).
Remark -When the maps F and G are both Marchaud and A-Lipschitz with respect to x, y, we deduce that U, satisfies

Decomposable Case
We shall consider first the decomposable case for which we have explicit formulas, that we next use to solve the general problem of finding a contingent solution to the problem If u: X H Y, we set When G is Lipschitz with nonempty closed images, we denote by llGlla its Lipschitz constant, the smallest of the constants 1 satisfying where B is the unit ball. For time-dependent set-valued maps G(t, .) which are uniformly Lipschitz, we still set llGlla := G(t, .) to denote the common Lipschitz constant.
Let @ : R+ x X I . , X and : R+ x X I . , Y be set-valued maps.
Consider the decomposable system of hyperbolic partial differential inclusions and its associated characteristic system of differential inclusions We denote by SLo(x) the set of solutions x(.) to the differential inclusion xt(s) E -@(ts, x(s)) on [0, t] starting at x.
Define the set-valued map U, : R+ x X I . , Y by7 We set eat -1 eatat -1 ey(t) := -& e;(t) := a a2 Theorem 4.1 Assume that @ : R+ x X -X and Q : R+ x X I . , Y are Marchaud maps and that Uo is closed with linear growth. Then the set-valued map U, : R+ x X I . , Y defined by (14) is the solution defined by the method of characteristics, and is thus the largest solution to (12) satisfying the initial condition  € U,(t, x), there exist a solution x(.) E S'*(x, .) to the differential inclusion x'(s) E -@(ts, x(s)) starting at x, ut E Uo(x(t)) and z(s) E q(s, x(ts)) such that If there ezist positive constants a, 6, Po, yo, P, 7 such that then Moreover, if Uo, @, Q are A-Lipschitz with respect to x, then the maps Ua(t, .) : X -+ Y are also Lipschitz (with nonempty values): We recall that Ua(t, -) being the solution defined by the method of characteristics, it is both a forward and backward solution to (12): Th' is means that it satisfies (12) and v (t, X, Y) E Graph(U), 0 E DU(t, x, y)(-1, -@(t, x)) + Q(t, x) (16) Formula (14) shows also under mere inspection that the graph of Ua(t, .) is convex (respectively Ua(t, a) is a closed convex process) whenever the graphs of the set-valued maps Uo, @(t, -) and Q(t, .) are convex (respectively @(t, .) and Q(t, .) are closed convex processes).

Proof
1. -We prove first that the map Ua is the largest solution to inclusion (12), i.e., that the tube Graph(Ua(t, .)) is the reachable map RGraph(uo,(t).
Indeed, a pair (x, y) belongs to RGraph(u (t) if and only if there exist 0) solutions (z(.), y(.)) to the characteristic system (13) starting from the graph of Uo and satisfying (z(t), y(t)) = (x, y). This solution can be written in the form I z(t) = wt + Ji @(s, Z(S))~S where ut E Uo(wt). By setting x(s) := z(t-s), we observe that it is a solution x(.) E SLO(x) to the differential inclusion xf(s) E -@(ts, x(s)) starting at x and such that wt = x(t) and that Hence this solution Ua coincides with the largest solution.
2. We prove now a comparison result between solutions to two decomposable partial differential inclusions.

Single-Valued Lipschitz Contingent Solutions
We shall now prove the local existence of a (contingent) single-valued solution to  We denote by T2(a) the smallest positive root of the equation when a is large enough for such a root to exist. Let T := min(Tl (p), T2(a)).
We infer that by Proposition 4.3 because u is of the form r(uO, ( P, , $,). Set T := min(Tl(p), T2(a)) and let us denote by B;(p, a) the subset defined by which is compact (for the compact convergence topology) thanks to Ascoli's Theorem.
We have therefore proved that the set-valued map R sends the compact subset BL(p, a) to itself.
It is obvious that the values of R are convex. Kakutani's Fixed-Point Theorem implies the existence of a fixed point u E R(u) if we prove that the graph of R is closed.
Actually, the graph of R is compact. Indeed, let us consider any sequence (v,, u,) E Graph(R). Since BL(p,a) is compact, a subsequence (again denoted by) (v,, u,) converges to some function But there exist bounded Lipschitz selections $, E G,, with Lipschitz constant ~llGI/~(l+ a) such that Therefore a subsequence (again denoted by) $, converges to some function $ E G,. Since cp, , converges obviously to cp,, we infer that u, converges to l?(uo,cpv, $) where $ E G,, i.e., that u E R(v), since l ? is continuous by Proof -Let u be any bounded single-valued contingent solution to inclusion (20). One can show that u can be written in the form t ~(t, x) = ut + Z(S)~S where z(s) E G(s, x(ts), u(x(ts))) by using the same arguments as in the first part of the proof of Theorem 4.1.