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A nonmonotone differential inclusion
NonlinearAnalysis, Theory. Methods & Applications, Vol. 11, No. 7, pp. 757-776, 1987.
0362-546X/87 $3.00 + .00 Pergamon Journals Ltd.
Printed in Great Britain.
A NONMONOTONE
DIFFERENTIAL
INCLUSION
P. LABORDE Laboratoire associ6 au C N R S No. 40226, Universit6 de Bordeaux I, 351, cours de la Lib6ration, 33405 Talence Cedex, France
(Received 25 November 1985; received for publication 11 April 1986) Key words and phrases: Differential inclusions, n o n m o n o t o n e set-valued map, discretization, uniqueness, asymptotic behavior. INTRODUCTION
LET E BE an Euclidean space, K a nonempty closed subset of E and ~p a convex function in E. Consider A the set-valued map with domain K vanishing in the interior of K and such that, for each boundary point v, Av is the set of outward normals to the level surface of ~p passing through v. We want to study the following differential inclusion in E, with state constraint du
d--t(t) + Au(t) ~ f(t),
u(t) E K,
(0.1)
where f is a given function in [0, T]. This problem arises in a preliminary study of models in theory of plasticity with "nonassociated laws", cf [10, 9]. We introduce a compatibility condition between K and ~p which is essential in this work. A special case, where this condition is satisfied, occurs when K is a convex set and 7Ythe gauge function of K. Hence Av is the normal cone to K at v, so A is maximal monotone, and (0.1) defines a sweeping process in the sense of Moreau [12]. In general, the operator A does not verify a monotonicity-type property. Available literature concerning nonmonotone differential inclusions essentially treats existence in a framework different from this article, cf., for example, [2]. We first prove the existence and uniqueness of a solution w to the inclusion
w+Aw~e,
wEK,
for a given element e in E. This result allows us to define an implicit discretization method for the differential inclusion. We show the convergence of this approximation, which yields the existence of a function u(t) verifying (0.1) a.e. in (0, T) and an initial condition. Under additional assumptions, we obtain a characterization of the velocity du/dt; moreover the uniqueness of solution u(t) is proved. Finally, we give a result concerning the asymptotic behavior of trajectories. The basic assumptions of this work lead to more technical proofs than in the monotone case. 1. T H E
SET-VALUED
MAP A
1.1. Definition and assumptions Let E be a real Euclidean space with the inner product denoted by (., • ) and the associated norm
I1"II757
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Notation. Int X, bd X and cl X are respectively the interior, the boundary and the closure of a subset X of E, respectively. We associate with a convex cone X, its polar X ° -{~ E E : (~, v) ~< 0 Vv ~ X}. If X is a closed convex set, N ( X ; v) denotes the normal cone to X at a point v : N ( X ; v) = (X - v) °, gx the gauge function of X : gx(v) = inf{r > 0 : v E rX}, and proj(X; v) the (normal) projection on X of a point v. The subdifferential of a convex function q9 at a point v is denoted by Oqg(v), and inf q0 stands for the infimum of function q0 on its domain. With these notations, let K be a nonempty closed subset of E, and ~p a convex function in E such that int(dom ~p) D K. Definition. W e call A the set-valued map from E to E, with domain K such that: if v ~ int K : A v = {0}, if v E bd K : A v is the normal cone to the level set {w : ~p(w) ~< ~p(v)} of ~p through v. Remark. Let v be a boundary point of K such that ~p(v) > inf lp. Then, we have (cf. [13, p. 222]): A v = {X : )C = ~ with ). I> 0 and ~ E O~p(v)},
(1.1)
i.e. A v is the closed convex cone generated by O~p(v). Now, we introduce a compatibility condition between K and ~p which is essential in this work; (1.2c) is the main property. Basic assumptions. We first suppose that the minimum set of ~p is nonempty and included in K: O 4: {v : ~p(v) = inf ~p} C K.
(1.2a)
Moreover, if v E bd K, the cone v + A v (with vertex v) must be pointing strictly outward to K, in the following sense: there exists a constant o~ ~ (0, 1] such that: X E A v implies (X, w - v) ~< (1 - ~)IlXllIIw - vii
/
(1 .2b)
Vw ~ g .
Finally, if v, w ~ bd K, the cones v + A v and w + A w must "diverge"; more precisely: v, w E K, w - v q~ ( A v ) °" imply (p, ~)/> 0
V~ E A w .
(1.2c)
p = proj((Av)°; w - v) 1.2. Examples 1.2.1. Let us suppose that K is a closed convex subset of E, containing the origin in its interior, and ~p is the gauge function g r of K. Then, for each v, A v is the normal cone N(K; v) to K at v. In other words, A is the subdifferential of the indicator function of K. Assumptions (1.2) are obviously satisfied (with tr = 1 in (1.2b)) a n d A is maximal monotone. In the following, this will be referred to as the classical case.
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759
1.2.2. Let ~p be the norm of E, and K a compact set containing the origin in its interior. So we have, for all boundary points v of K (cf (1.1)): A v = {x : X = ~ v , ~ >~O}.
Suppose that the gauge function of K is Lipschitz-continuous on the unity sphere of E. (In particular, this holds when K is convex.) Then assumptions (1.2) are fulfilled, cf. [8]. But A is not a Lipschitz perturbation of a maximal monotone set-valued map in general. 1.2.3. In the previous example, the function ~p is differentiable on the boundary of K. As a consequence, when v lies on bd K, A v is reduced to a half-line. We now consider a case where ~p is not differentiable. Let E be the space of symmetric endomorphisms of ~", equipped with its Euclidean structure. Denote sp(v) the set of eigenvalues of an endomorphism v, and set: 2,/~sp(v)
~p(v) = max{). - / ~ : 2,/~ ~ sp(v)}
Vv E E.
Then, the corresponding set-valued map A obeys conditions (1.2); cf. [9, 10]. 1.2.4. Finally, let us give two examples in E = R 2. In the first case the domain K of A is not convex:
g = {v = {Or, v2} ~ ~ :
02 - I o l l < 0, o21 + v~ < 2},
~p(v) = v~ + (v2 + 1) 2
Vv ~ 4 2.
In the second example, the minimal set of ~p intersects the boundary of K: K = {v E R2: vt <~ 0, vl + Iv21 ~<0},
~p(v) = max{Iv21, max{v l, 0}}. The basic assumptions are verified in each of these two cases. 2. F I R S T P R O P E R T I E S
We now deduce from the previous basic assumptions some useful properties of A. 2.1. Characterization o f A
Let cr be a continuous function from E to the interval [inf ~p, +oo) such that o(v) {~ ~p(v)
ifv E b d K
~p(v)
ifv~intK,
<~V(v)
ifv~K.
Such a function a always exists; we will give an example. Now, define for each v in E: Q(v) = {w: ~p(w) ~< or(v)}
which is a nonempty closed convex subset of E. The following characterization of A holds.
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LEMMA 1. Under assumption (1.2a) the condition v E K is equivalent to v ~ Q(v); furthermore
A v = (Q(v) - v) °
for all r i n K .
The proof is easy. In example 1.2.1, where ~p is the gauge function of the convex set K, it is enough to take tr(o) = 1, i.e. Q(v) = K, for all v. In order to construct a function a into the general case, we associate with the set K a continuous function q~ such that {:1 tp(v)
ifv E b d K 1 ifvEintK,
(2.1) >1
ifvq!K.
One may take
qg(v) = 1 + e(v) dist(bd K; v), with e(o) = 1 if 0 $ K, - 1 if v E K, and where dist(X; w) denotes distance of a point w from a subset X in E. Naturally, when K is convex we may also choose tp = gK. Then, let:
a(v) = max{~(v) + c(1 - qg(v)), inf %0}
Vv ~ E,
with c > 0. According to (1.2.a), this function tr works.
2.2. A continuity property The previous characterization is a convenient tool to establish a result of closed graph in the functional space
{
L 2 ( O , T ; E ) = v:(O,T)--->E,
So
}
[Iv(t)ll2dt< + oo ,
equipped with its Hilbertian structure. THEOREM 1. Assume (1.2.a) and let u,, %, be two sequences in L2(0, T; E) such that:
(i)x,(t)~Au,(t) a.e. t E (0, T), Vn; (ii) u, --->u in L2(0, T; E), u being a continuous function; (iii) %, ~ %weakly in L2(0, T; E). Then, x(t) E Au(t) a.e. t E (0, T). (In particular, the graph of A is closed in E × E.) In the classical case, A = N ( K ; . ) is a maximal monotone operator, and the closeness property of the theorem is well-known, cf. [4].
Proof. Let Q be the set-valued map defined at v E L2(0, T; E) as (cf. Section 2.1) a ( o ) = {w : w E L2(0, T; E), w(t) ~ a ( o ( t ) ) a.e. t}. First, let us verify that Q is lower semi-continuous (l.s.c.) in L2(0, T; E). Precisely, suppose that a sequence u, converges to u in L2(0, T; E), and let v ~ L2(0, T; E) be verify v(t) E Q(u(t)) a.e.t.
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761
The question is whether there exists a sequence on tending to v in L2(0, T; E) such that v,(t) ~ Q ( u , ( t ) ) a.e.t. But, let on(t) = p r o j ( Q ( u n ( t ) ) ; v(t)), a.e.t. The continuity of the function tr implies the continuity of the map{w, z}---> proj(Q(w); z) from E × E to E. Hence, owing to the dominated convergence theorem, the sequence vn converges in L2(0, T; E) to proj(Q(u); v) which is equal to v. Thus Q is l.s.c. Let us make use of conditions listed in theorem 1. The 1.s.c. of ~), and the characterization of A (lemma 1) applied to Zn U A u , , yield: oT (g(t), v(t) - u(t) ) dt <<-O.
We deduce the pointwise inequality: (Z(O, v(t) - u(t)) <- 0
a.e. t,
Vv ~ Q(u(O),
due to the continuity assumption on u and a standard theorem of integration theory. The conclusion of the theorem follows from the characterization of A. 2.3. Pseudo-distance associated with A Let us denote, for all ul and u2 in K:
p~ = proj((Aui)°; uj - ui)(] ~ i)
(2.2a)
dA(ul ;u2) = min{llp~ II, ILOzlI}.
(2.2b)
In the classical case, we find: d,a(ul, u2) = Ilul - u211 for all ul, u: ~ g.
In general, the quantity dA(Ul, u2) gives an estimate of the distance between two elements ul and u2 of K in the following way. Prtoposrnon 1. Under assumption (1.2) we have the two inequalities
/211u - u211 d a ( u l , u2)
Ilu - u211
Vua, u2
K,
where a, is the constant of assumption (1.2.b). Proof. T w o cases may happen: either
(a)
u2 - ul E (Aux) ° and ul - u2 E ( A u : ) °,
or (with an eventual permutation of indices)
(b)
u2 - ul
(AuO °.
In case (a), we have dA(ul, u2) = I}ul -- u21}, and the two inequalities hold. Examine the second case. First, check the implication below: u2 - ul q~ ( A u l ) ° ~ u l - u2 E (Au:) °.
(2.3)
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Precisely, by definition of A, the condition (u2 - ul) qi_ (Aul) ° implies that u I ~ bd K and ~P(ul) < lP(u2), whence we deduce that (Ux - u2) ~ (Au2) °. It follows that Ilul - uzll z --Iho ll z.
(2.4)
Now, recall the result of orthogonal decomposition according to mutually polar cones. Let C be a nonempty closed convex cone in E and w ~ E; then (cf. [7]). ~ w = p + q with p = proj(C; w) ¢:> t{p, q} ~ C x C O and (p, q) = 0.
(2.5)
By decomposition of u2 - Ul according to A u l and (Aul) °, assumption (1.2b) yields to ]lu2 - uall 2 1> [Lolll2 ~ og[lu1 - - U2[)2.
(2.6)
The proposition is a consequence of (2.4) and (2.6). 3. SOLUTION OF SOME TIME-INDEPENDENT INCLUSION In this section we study the solution w of the inclusion w + Aw ~ e
(w E K)
(3.1)
for an element e given in E. We shall also give a tangential condition satisfied by A and connected with the study of inclusion (3.1). 3.1. Projection associated with A THEOREM 2. Suppose (1.2) holds and let e in E. There exists a unique u lying in K solution of inclusion (3.1). If e I and ez are any two elements of E, let ui be the solution of ui + Aui ~ e~ (i = 1, 2); then IlUl - u2112
-111el - e2LI
where c~ is the constant of assumption (1.2b). Definition. The projection associated with the set-valued map A is the map PA : e---~ u such that u + A u ~ e. Owing to theorem 2, PA is Lipschitz-continuous in E with the Lipschitz constant L -- c~-1/2. This definition is motivated by lemma 2 hereafter and also by the following remark. Remark. The map PA corresponding to the classical case (Section 1.2.1) is the normal projection on the convex set K. The Lipschitz constant of PA given by the theorem is L -- 1 in this case; that is the classical contraction property of proj(K; • ). Proof of the theorem. If e E K, a trivial solution of inclusion (3.1) is u = e. Suppose that e K and associate with K a continuous function q0 : E--~ R verifying (2.1). Let w(r) be the projection of e on the level set {z : , ( z ) ~< r}. According to assumption (1.2a), the minimizer w(inf ~p) of function ap lies in K; thus cp(w(inf ~p)) ~< 1. Moreover, since e 4= K, we also have qg(w(~V(e)) > 1. A continuity argument ensures the existence of f ~ [inf ~p, ~p(e)) such that q0(w(r--)) -- 1, i.e.
A n o n m o n o t o n e differential inclusion
763
W(r-) is a boundary point of K. Then, the definition of w(r) implies that (e - w(r-)) E Aw(r-), i.e. u = w(r-) is a solution to inclusion (3.1). Now, let el, e2 E E and ui be a solution of ui + Aui ~ ei (i = 1, 2). We use the decomposition in cases (a) and (b) of the proof of proposition 1. In case (a), one has: ( ( e I -- U l ) -- ( e 2 -- u 2 ) , u 1 -- u 2 ) ~ 0 , SO
IlUl - u ll
lie1 - e211-
Let us consider case (b). By hypothesis (1.2c) the elementpt = proj((AUl)°; u2 - ul) verifies: (p,, (e2 - u2) - (el - u l ) ) / > 0. Then Ilpl]l2 ( = (p,, u2 - u,)) is less than (Pl, e2 - el), i.e. ]IP,II ~< lie1 -- e21], so (2.6) implies
Ilu, - u2ll
~
ot-1/Zlle~
-
e21l.
(3.2)
This inequality is also true in case (a). In this way we deduce the uniqueness of the solution u to u + A u ~ e and the Lipschitz property of the theorem. Moreover, we have LEMMA
2. The map PA verifies properties (i), (ii), (iii):
(i) p2 = PA, PA(E) = K; (ii) v E K ~ P A V = V ; V ~ K ~ P A V E (iii) X E A(PAV) ~ PA(V + X) = PAY;
bdK;
in particular:
v E K ~ P A ( V + A v ) = {v}. The proof of the lemma rests with the following facts (cf. [9]). First, A v contains the origin for all v in K. Second, A vanishes inside the interior of K. Finally, the values of A on the boundary of K are convex cones. 3.2. Boundary condition Let us denote by T(X; v) the contingent cone to a closed subset X of E at a point v. By definition, T(X; v) is the set of directions 0 such that there exist two sequences 0n and hn verifying (cf. [1]):
On--'>OinE, hn-'-~O, h n > O v + h n On ~ X
Vn,
Vn.
When X is convex, T(X; v) reduces to the tangent cone to K at v. We indicate a first geometrical property of A. LEMMA 3. Assumptions (1.2) o n A imply
(g - Ao) n T(K; v) 4 : 0
VvEK,
VgEE.
(3.3)
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Let us comment on this result. For a given e in E, put
F(v) = e - t) - At)
Vv E K.
A solution u of the inclusion u + A u ~ e is an equilibrium state for the set-valued map F, i.e. u verifies: F(u) ~ O. On the other hand, property (3.3) yields
F(v) n T(K; v) q~ 0
Vv E K.
(3.4)
This expression is the tangential condition which appears in results for equilibria of setvalued maps, cf. [2]. Nevertheless, the assumptions of continuity and compactness used in these references are not verified in our problem.
Proof of lemma 3. Let Uh = V + hg and Oh = P AUh. The Lipschitz-continuity of P A implies that the s e t { h - l ( t ) h -- O)}h>0 is bounded in E. Hence, we can extract a sequence converging to an element 0 in E. The condition oh E K ensures that 0 E T(K; v). Moreover: g - h - l ( V h - t))(= h - l ( U h -- Oh) ) E A t ) h . Since the graph of A is closed in E x E (cf. theorem 1), the convergence of Oh to V shows that: g - 0 E At). The lemma follows. Now, let us give a boundary condition simpler than (3.3) which will be used further. LEMMA 4. The condition:
( - A v ) O int T(K; v) ~ 0
Vv E K
(3.5)
implies (3.3). The converse is true when K is a convex set with a nonempty interior (in particular, assumption (1.2) and the convexity of K imply (3.5)).
Proof. First, suppose that (3.5) holds: there exists 0 E ( - A v ) O int T(K; v). Let g in E. The condition 0 E int T(K; v) implies the existence ofA > 0 such that (g + A0) E T(K; v). On the other hand, since 0 lies in cone ( - A v ) , one also has (g + ~,0) E g - Av. In this way we obtain (3.3). Conversely, suppose condition (3.3) and the convexity of K. The interior of K is nonempty, so there exists 0 such that T(K; v) + 0 C int T(K; v). Choosing g = - 0 in (3.3), we also have ( - A v ) n (T(K; v) + 0) g 0 . Property (3.5) follows. 4. A P P R O X I M A T I O N OF THE D I F F E R E N T I A L INCLUSION
We are now in a position to find a solution u(t) to the differential inclusion:
u'(t) + Au(t) ~ f(t)
(u(t) E K),
(4.1)
with the initial condition: u(0) = u0; the prime in (4.1) stands for the derivation with respect to t.
(4.2)
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765
Let us put conditions on data: f ~ L 2 ( 0 , T;E)
and
Uo ~ K .
(4.3)
4.1. Discretization Let k > 0 and N be the integer such that ( N - 1)k < T ~ Nk. Denote tn = nk for n = 0, 1. . . . , N - 1, tN = T and 1~ = (tn- 1, tn). Let also fk be a sequence of functions converging to f in L2(0, T; E), such that fk(t) = f ~ constant on In. The discretized problem is to find a continuous piecewise affine function Uk in [0, T] solution of (4.4) (4.5). U k ( t ) = k - l ( t ~ -- t)u~,-1 + k - l ( t - t n _ l ) u ~ k - l ( u ~ - u~ -1) + AuT, ~ fT,
Vt~I n
(u~ v= K)
(4.4a) (4.4b)
for n = 1 , . . . , N and u ° = u0.
(4.5)
With the definition introduced in Section 3.1, the implicit algorithm (4.4b) may be written: u~ = PA(u~ -1 + kf"k).
(4.6)
Remark. In the particular case A = N ( K; . ), the differential inclusion (4.1) describes a sweeping process by a mooing convex set, in the terminology of Moreau. Moreover, expression (4.6) defines the associated catching up algorithm of which the convergence is studied by this author in [12].
THEOREM 3. Suppose that hypotheses (1.2) and (4.3) are satisfied. Then, for any fixed k > 0, the discretized problem (4.4) (4.5) admits a unique solution uk. Furthermore, there exists a subsequence k(n) converging to zero and an absolutely continuous function u in [0, T], such that: Uk(,) ~ U uniformly, in [0, T], u~,(n)~ u'
weaklyin L2(O, T; E).
The limit u verifies the differential inclusion (4.1) a.e. in (0, T), and the initial condition (4.2). Finally, we have the estimate:
Ilu'(t)ll
zllf(Oll
a.e. t,
where L is the Lipschitz constant of PA. Remark. In [10] (cf. also [9]) is studied an approximation method of the differential inclusion by penalization-regularization. Note also that [3, 6] give others' existence results for differential inclusions in the same direction. Proof. The Lipschitz continuity of PA (theorem 2) and equality (4.6) imply:
IIk-l(u~ - u~-l)ll <~Zllf~ll
n = 1, 2 . . . . , N.
(4.7)
Since the sequence fk is bounded in L2(0, T; E), the same holds true with respect to the
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sequence u~,. Thus, one can extract a subsequence Uk(,) which converges uniformly in [0, T] to some absolutely continuous function u such that: u~,(,) ~ u' weakly in L2(0, T; E).
(4.8)
Theorem 1 allows us to pass to the limit in the discretized equation (4.4b). Therefore the function u verifies the differential equation (4.1.) a.e. in (0, T). Finally, we deduce from (4.7), (4.8) that [lu'(t)l[ ~< LIIf(t)ll a . e . t . 5. E X P R E S S I O N
OF THE VELOCITY
The purpose of this section is to express the derivative, at a time t, of a solution u to differential inclusion (4.1) in terms of the position u(t) and of the right-hand side f(t). Henceforth, we suppose in addition that K is a (closed) convex set.
(5.1)
Notations. For a function g, put g(t + O) =
lim
h--*0,h>0
h -~
i
t+h
g(s) ds,
when this limit exists, t is called a Lebesgue point from the right of g. Furthermore, we denote g' (t + 0) the derivative from the right of g at t, q0'(v; d) the directional derivative of a function q9 at a point o into a direction d, q9'(v) the gradient of function 99 at v, when these quantities exist. Finally, x + stands for max{x, 0}. Let us first study the differentiability properties of P3 at v such that v verifies either (a) or (b);
]
(a)
v ~ int K,
}
(b)
v E bd K and ~p is Gateaux-differentiable at point o.
(5.2)
Notice that case (5.2a) is trivial, since PA is equal to the identity on int K. If o verifies (5.2b) the gradient ~p'(v) is different from zero (otherwise A o = E, which contradicts assumption (1.2b); moreover A o -- ~+~p'(v) (cf. (1.1)), and the boundary condition of lemma 4 becomes -~p'(o) ~ int T(K; o).
(5.3)
PROPOSITION 2. Let o be such that (5.2) holds and d E E. There exists a unique element 0 which satisfies the equivalent conditions (i) and (ii): (i) 0 = d - / . ~ ' ( v ) with 1. = inf{tt >1 0 : d - # ~ ' ( v ) ~ T(K; v)} (ii)
d = 0 + • with 0 E T(K; v), X, E A v , if d C T(K; v) : X = O, if not : 0 ~ bd T(K; v).
Moreover, PA admits a directional derivative at v into direction d equal to P'A(O; d) = O.
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767
Proof. (a) Let us first verify that conditions (i) and (ii) define the same unique element 0. It is obvious in the case d E T(K; v) where 0 = d. Examine the opposite case: v E bd K and d q?. T(K; v). Let v(/0 = d - / ~ p ' ( v ) and A = {/u/> 0 : v(/~) ~ T(K; v)}. The boundary condition of lemma 3 ensures that A 4= O. Let ~, = infA; by definition, v(2,) is the (unique) element 0 which verifies condition (i). In fact v(~,) ~ T(K; v), due to the closeness of T(K; v); moreover ~, > 0, since v(0) $ T(K; v). The definition of ~, implies that v(/~) $ T(K; v) for all/~ @ [0, ).[, it follows, by continuity: v(;t) E bd T(K; v). Then, condition (ii) is satisfied by 0 = v(A), Finally, consider any element 0 verifying (ii). There exist v/> 0 such that 0 -- v(v). The convex cone T(K; v) has v(v) on its boundary and ( - ~ ' ( v ) ) in its interior. Consequently v > 0 and we have for all/t E [0, v[: r(/~)(= v(v) + (9 - ~)~p'(v)) ~ T(K; v). We conclude that t, = ~., i.e. v(~.) is the unique element 0 which obeys condition (ii). (b) We now study the differentiability of PA at a point v verifying (5.2) into a direction d in E. Denote Wh = PA(V +hd). First, we suppose: d ~ int T(K; v). The Lipschitz-continuity of PA implies that
IIh-l(wh
-
v)lt < Zlldll
Vh > 0.
Then, there exists a subsequence h(n) ~ 0 and v E E such that h(n)-l(Wh(n) -- v) ~ "r. But we have, by definition of Wh:
h-l(Wh -- V) ÷ A w h ~ d. Owing to the closeness of the graph of A (cf. theorem 1), it follows that v + A v 3 d. Furthermore v E bd T(K; v), since d ~ int T(K; v), i.e. w h ~ bd K for all h > 0 (lemma 2, (ii)). Then the limit v is equal to the element 0 given by condition (ii) of the proposition; and the whole family h-l(Wh -- l)) converges to 0. The same holds true when d E int(K; v), because we have for all h positive and small enough: v + hd E K, so Wh = V + h d (lemma 2, (ii)), i.e. h-l(Wh -- I)) = d. In conclusion, for any d in E, the directional derivative P'A(V; d) exists and is equal to the element 0 above defined. That ends the demonstration. We easily deduce from this proposition that: COROLLARY 1. Let v be a boundary point of K such that ~p and the gauge function gr are Gateaux-differentiable at v; denote: and
v(v) = kg'K(V),
where k is positive and such that (v(v), r/(v)) = 1. Then, there exist directional derivatives of PA at v and
(P'A)(v; d) = d - (d, v(v))+r/(v)
foralldinE.
If v is an interior point of K, we have:
(PA)'(V; d) = d
for alld.
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The study of the differentiability of Pa enables us to express the velocity of the solution u(t) of differential inclusion (4.1). THEOREM 4. Assume (1.2) and (5.1) hold. Let u be an absolutely continuous function in [0, T] solution of: u' + A u ~ f a.e. in (0, T) (theorem 3). Let also to E [0, T) be a Lebesgue point from the right of f, such that v = U(to) verifies condition (5.2). Then, the function u has a derivative from the right at time to given by
u'(to + O) = (PA)'(U(to);f(to + 0)). Remark. This theorem and the previous proposition are to be connected with the following results known in the classical case. First, the normal projection projr on the closed convex set K possesses directional derivatives at any point v E K and (cf. [14, 11]): ( p r o j r ) ' ( v ; d) = proj(T(K; v); d)
Vd E E.
Furthermore, by application of a theorem of Brrzis (cf. [4]) to the case A -- N(K; • ), one has
u'(to + 0) = proj(T(K; U(to));f(to + 0)), for any Lebesgue point from the right to ~ [0, T) of f. To prove theorem we shall need a preliminary property. LEMMA 5. Let h(n) be a sequence of strictly positive numbers tending to zero such that U(to + h(n)) ~ bd K for all n. Then, there exists lim h(n)-l[U(to + h(n)) - u(t0) ] = (P'a)'(U(to);f(to + 0)). n
Proof. Let us denote, for all h G (0, T - to): Oh =
h-l[U(to + h) - U(t0)],fh = h -1
f
to+h
f(t) dt,
to
Xh = h-1
f
to+h
X(t) dt
where X = f -
u' a.e.
t0
Moreover, if X is a subset of E and/3 E (0, 1), denote by Xt~ the cone X¢ = {~ E E : there exists g E X s u c h that (X, ~) I> (1 -/3)Ilxll I1~11}. With these notations, the mean-value equation derived from (4.1) is written Oh ÷ Xh ----fh"
(5.4)
On the other hand, the solution u(t) is a continuous function and X(t) E Au(t) a.e.t. From the upper-semi-continuity of subdifferential 0~p (cf. [13]) we deduce, for all/3 > 0, the existence of ho(/3) > 0 such that
g(t) E (Zu(to) ) ~ a.e. t E (to, to + ho(fl) ).
A nonmonotone differentialinclusion
769
The convexity of (Au(to))~ and the mean-value theorem yield:
Xh E (Au(to))13
Vh E (0, ho(fl)) (Vfl > 0).
(5.5)
Now, let h(n) be a sequence tending to zero, such that U(to + h(n)) E bd K for all n. In a first step we suppose that U(to + h(n)) 4: U(to) for all n. Owing to condition (5.3), there exists fl > 0 such that
-(Au(to))31j C T(K; U(to) ). The sequence of boundary points U(to + h(n)) tending to u(t0), it follows that
Vh(,) q~ -(Au(to))2#
Vn >I nl,
(5.6)
for some integer nt. On the other hand, according to (5.5), there exists another integer n2 such that:
Xh(,) E (Au(to))t3
Vn >~n2.
(5.7)
This last property, together with (5.6) and the mean-value equation (5.4), implies the existence of a positive constant 7 such that:
[Ivh(.)l[ +
Ilxh(.)ll fllA(.)ll
Vn 1> max{hi, n2}.
One deduces that sequences Oh(,) and ~Ph(,) are bounded, and there exist subsequences such that
Vh(,') ~ Vo,
gh(,') ~ gO.
We derive from (5.4) that v0 + X0 = f(to + 0). The hypothesis U(to + h(n)) E bd K for all n leads to v0 E bd T(K; U(to)). Passing to the limit as fl---~ 0 in (5.7), we also obtain X0 E Au(to). It follows that v0 = Pa(u(to);f(to + 0)) by proposition 2 and the whole sequence Vh(,) tends to v0. In a second step, let us study the case where U(to + h(n')) = U(to) for a subsequence h(n') of h(n). Sincefh tends tof(t0 + 0) as h ~ 0, expressions (5.4) (5.5) imply thatf(t0 + 0) E Au(to), so (Pa)'(U(to);f(to + 0)) = 0 due to proposition 2. Using the result of the first step, we deduce that the whole sequence vh(,) tends to 0. The proof is complete.
Proof of the theorem. We keep the notations of the previous lemma. First, consider the case where: there exists h0 > 0 such that: u(t) ~ int K
Vt E (to, to + h0).
(5.8)
Since Z(t) = 0 a.e. t ~ (to, to + h0), equation (5.4) becomes: Vh =fh for all h in (0, h0). Then f(to + O) E T(K; U(to)), and there exists u'(to + 0) equal to f(to + 0). By proposition 2, we obtain the conclusion of theorem when (5.8) holds. Let us now examine the opposite case. We consider any sequence of numbers k(n) which converges to zero, and we define
h(n) = sup{h E [0, k(n)] : U(to + h) ~ bd K}.
770
P. LABORDE
The negation of (5.8) ensures that h(n) > 0. Since u(t) E int K for all t lying in the interval (to + h(n), to + k(n)), we have X(t) = 0 a.e. t in this interval. By definition Of%h, it follows that
h(n)
Xk(,) = k - ~ %h(,).
(5.9)
As U(to + h(n)) E bd K for all n, we can use the previous lemma for sequence h(n). Then, the sequence %h(,) tends to Z0aP'(u(t0)) with (proposition 2): ~.0 = i n f ~ / > 0 :f(t0 + 0) - #~p'(U(to)) E T(K; u(t0))}. So is the sequence Zk(,), due to (5.9). Consequently ok(,) converges to (Pa)'(u(to);f(to + 0)); this ends the proof. 6. A UNIQUENESS RESULT The uniqueness of the solution of the differential inclusion is now established under some
regularity condition on the multi-valued map A; we also suppose that the right-hand side f of the differential inclusion is piecewise constant. In the classical case, the uniqueness results from the well-known property of decrease of the distance between two trajectories with respect to time, due to the monotonicity of A -- N(K; .). In our problem, we show that the pseudodistance da (cf. Section 2.3) between two possible solutions satisfies a Gronwall condition. Assume that: ~p is a %2-function in bd K
(6.1)
{v : ~p(v) = inf ~p} C int K.
(6.2)
THEOREM 5. Suppose (1.2), (5.1), (6.1), (6.2) and let f E E, u0 E K. There exists a unique Lipschitz-continuous function u in [u, + ~ ) such that:
u'(t) + Au(t) ~ f
a.e. t > 0; u(0) = u0.
(6.3)
A straightforward consequence is: COROLLARY 2. If f is a piecewise constant function in [0, T], the solution of u' + Au ~ f a.e. in (0, T), and u(0) = u0, .o unique. The Lipschitz-continuity of a solution, in theorem 5, is deduced from theorem 3. The uniqueness result is based on two lemmas hereafter. First, trajectories for the differential inclusion with a constant right-hand side behave as it follows. LEMMA 6. Let u be a Lipschitz-continuous function in [0, + ~ ) satisfying: u'(t) + A u(t) ~ f a.e. t > 0. These exists a time to E [0, +oo] such that u(t) E int K when t ~ (0, to), and u(t) E bd K when t/> to.
Proof. We eliminate the trivial case: u(t) E int K for all t > 0, which corresponds to to -- +oo in terms of the lemma. Consider the opposite case and define to --- inf{t > 0 : u(t) E bd K}.
A nonmonotone differential inclusion
771
The question is whether the trajectory u(t) lies on the boundary of K for all t I> to. We proceed by contradiction. So, suppose the existence of a time t2 > to such that u(t2) E int K, and let t 1 = sup{t <~ t 2 : u(t) E bd K}. Necessarily we have tl E [to, t2); moreover t~ > 0 since tl = to implies to > 0. Let us consider the differential inclusion (6.3) and set
X(t)=f-u'(t)
a.e. t > 0.
By an integration it follows, for all t3 1> 0 (t3 4: tt):
(
~"
u(t3) t3
u(tl).~ + _ _ tl
/
t3
(u, X(t)) dt = ( u , f ) ,
(6.4)
tl
where u is a given nonzero vector normal to K at U(tl). On the one hand, take in this equality t 3 = t 2. Since g = 0 a.e. in (q, t2), and u(t2) ~ int K, we obtain the strict inequality ( v , f ) < O. On the other hand, the boundary condition (3.5) at the point v = U(tl) and the continuity of derivative ~p' at this same point, prove the existence of some time t 3 E [0, tl) such that (v, X) ~> 0 a.e. in (t3, tl). Equality (6.4), for this choice of t3, yields: 0 ' , f ) ~> 0. This implies a contradiction, and the lemma holds. Now, the regularity assumptions (6.1), (6.2) enables us to prove that: LEMMA 7. Let u~ and/-/2 be two Lipschitz continuous functions in [0, + ~ ) such that (i = 1, 2)
u[ + Aui ~ f
a.e. in (0, +oo),
ui(O) = Uo.
Let also
r(t) ----dA(Ul(t),
u2(t)) 2
(cf. (2.2)). Then, the function r is locally Lipschitz-continuous in [0, +oo). Moreover, for any T < +0% there exists a constant C such that:
r'(t) <~ Cr(t)
a.e. t E (0, T).
Proof. (a) Taking into account the equality u[ = f a.e. in {t > 0: ui(t) E int K}, we deduce from lemma 6 the existence of to E [0, +oo] such that: ul(t) = u2(t ) ul(t)
and
V t ~ [0, to],
u2(t) E b d K
Vt>1 to.
Whenever to = +oo we have Ul(t ) = u2(t ) for all t/> 0, and the lemma is obviously verified. So suppose that to < + ~ . By definition, we have r = min{llPlll 2, liP2112}with (j 4= i)
pi=(uj--ui)-(uj-ui,
Tli)+~li; ?li=][~tY(ui)ll-llp'(ui)
t<~to .
772
P. LABORDE
Let T E (to, +oo); the function ~i is Lipschitz-continuous in [to, T], due to the regularity of trajectory ui and hypotheses (6.1), (6.2). Precisely:
I1~;11 ~ c, (T)
a.e. in(to, T),
(6.5)
where C(T) is a constant defined by the Lipschitz constant of ui(t) and by M and ro such that M =
sup{llw"(u~(/))ll,
t ~ [to, T]} < + ~ ,
ro = inf{llw'(ug(t))[I, t ~ I/o, T]} > 0. Thus the function IIp,II2 is locally Lipschitz-continuous in [to, +~), and the same holds true with regard to r (cf. [5]). Since d (pi,~i)-~t(u.i-ui,~i)+=O
a.e. t ~> to,
the derivative of r is d
2
a.e. in {t/> to : Ilpi(t)ll ~ Ilpj(t)ll(J * i)}
(6.6a)
ld 2 dt Ilpill2 = (pi, u] - u~) - (rli, uj - u i ) + ( p i , rl[ ).
(6.6b)
r' = ~ IlPill with
(b) Let us prove that function r verifies the Gronwall's condition. Denote (i = 1, 2) Ji = {t >I to : uj(t) - ui(t) ~. ( A u i ( t ) ) ° Io = Jt f'l J2,
(j ~ i)},
Ii = [to, +oo)\Ji.
The set Ji is closed, by continuity of functions Uk(t) and ~p'(v). Moreover, we deduce from (2.3) that J1 O J2 = [to, +oo); that is the decomposition I0, 11, 12 defines a partition of [to, +oo). In Io, we have: Ilplll2 = lip2112= Ilul - uzl]2. It follows from (6.6) that: d r'=~llux-u2112~-(aul-Au2,ul-u2)
a.e. in/0,
hence r' ~ 0 a.e. in Io. In Ix, where r = Ilplll2, we have simultaneously:
llp111~ ~> ~lu~
-
u~ll ~
(cf. (2.6)), and (Pl,U~-U[)E-(Pl,Au2-Aua)<~O
a.e. in11
(due to assumption (1.2c)). For a fixed T in [to, +oo), these facts together with (6.5) and (6.6) imply the existence of a constant C such that d
r' = ~ IIp,II 2 -< Cl[p,II ~ -- Cr
a.e. in I1 t~ (0, T).
A nonmonotone differential inclusion
773
The situation is symmetrical in 12, and the lemma follows.
Proof of the theorem. The previous lemma shows that the function r verifies the Gronwall's condition. So, proposition 1 implies that Ul = u2. 7. A S Y M P T O T I C
BEHAVIOR
7.1. Equilibrium Let f b e a nonzero vector. An equilibrium ~ for the differential inclusion u'(t) + Au(t) ~ f is a solution of the inclusion A l / D f (t/E K). We strengthen assumption (1.2c) by condition (7.1): ~p is strongly convex (i.e. there exists Y > 0 such that: ~0(Zu + (1 - ;0v) ~< Z~p(u) + (1 - ).)~p(v) - Z(). - 1)~[u -
vii2/
(7.1a)
for all u, v in E and 1. in [0, 1]); there exists a positive constant p such that we have:
w - v q~ (Ao) °
}
I(P' X) >~Pllw - oll Ilpll Ilxll
/ (7.1b)
p = proj((mv)°; w - v) - implies~for allz ~ aw. Moreover, assume that K is a compact set.
(7.2)
In the classical case, an equilibrium r/is a contact point of K with the supporting hyperplan to K normal to f, and assumptions (7.1), (7.2) are equivalent to the strong monotonicity of 0gr. In the general case we now prove the following existence result. THEOREM 6. Under hypotheses (1.2a, b), (7.1), (7.2), there exists a unique equilibrium ti.
Proof. (a) Existence. We use a procedure similar to that in the proof of theorem 2. Let r ~> inf ~p and w(r) which maximizes v ~ (/~, o) subject to lp(v)~< r. Owing to the strong convexity of ~p, we define a unique element w(r), and the map r--* w(r) is continuous in [inf ~p, +~). The compactness of K implies that r0 = sup{~O(v) : v E K} < +~. Let tp be a continuous function associated with K by condition (2.1). It follows that q~(w(inf ~p)) <~ 1 ~< q~(w(ro)). There exists ? such that tp(w(?)) = 1, by continuity; so we have: Aw(f) ~f. (b) Uniqueness. Let ut and u2 be two equilibriums. We again use the decomposition into cases (a) and (b) of the proof of proposition 1. In case (a), the equality ut --- u2 is provided by the strong convexity of ~p. Let us consider case (b) and define Pl = proj((Aux)°; u2
- Ul).
774
P. LABORDE
On the one hand, the condition )~E Au2, together with assumption (7.1b) and property (2.6), implies that ( P l , f ) ~> plllUl - u2112,
p , > 0.
On the other hand, since ]~E AUl, we also have ( P l , f ) ~< 0. It results that Ul = u2.
7.2. Behavior of trajectories Let us consider a trajectory u(t), that is a Lipschitz-continuous function such that u'(t)+Au(t)~f
a.e.t>0.
We introduce a map ~ which will play the role of a Liapunov function for this problem. Denote Pv the (normal) projection on the hyperplan orthogonal to the gradient ~p'(v), i.e. Poz
( n ( v ) = IIw'(v)ll-X~,'(v)).
= z -
(z,
~(v))~(v)
T h e n w e define ~e as ~e(v) -- ½11e~(v - a)ll z
for all v ,
where a is the equilibrium, cf. Section 7.1. We require further regularity conditions, first on ~p with assumption (6.1), and next on K: the gauge function q9 of K is convex and twice Frechet-differentiable.
(7.3)
THEOREM 7. Assume (1.2a, b), (6.1), (6.2). (7.1), (7.2), (7.3) hold. Let a be an equilibrium and u(t) a trajectory. There exists a neighbourhood N of a so that, if U(tl) lies in N for some tl >t 0, we have
~(u(t)) ~ O,
u(t) ~ a
when t----~ +~.
Remark. Aubin [1] introduces a concept of U-monotone maps which enables him to construct Liapunov functions associated with set-valued maps. This author derives some results of asymptotic behavior for differential inclusions. The notion of U-monotome map is not suitable for our purpose and this is why condition (7.1) is introduced. We need two preliminary results in order to prove theorem 7. LEMMA 8. The map ~ is convex and Frachet-differentiable in E. Moreover there exists a neighbourhood No of a and a constant fl in (0, 1) such that
~e(v) I>/~llv -
all 2 for all v ~ (bd K) M No.
Proof. The convexity and the differentiability of ~ come out from the linearity of Pa. The differentiability of the gauge function q9 implies (q~'(a), v - a) = Ilv - all
e(v - a)
when v lies in bd K, e(0) tending to zero together with II011. Using the notations of corollary 1, we obtain: ( 0 ( a ) , v - a ) z ~ IIv - a l l 2 ( a + e ( o - a ) )
A nonmonotone differentialinclusion
775
for all boundary points v. In this relation the constant 6, namely 1 - (r/(ti), v(ti)) 2, is strictly less than 1 by assumption (1.2b). The lemma follows. Let to >/0 be the time such that u(t) E int K when t E (0, to) and u(t) E bd K when t t> to (cf. lemma 6). The compactness of K ensures that to is finite when f:/= 0. LEMMA 9. The function t---->~£(u(t)) is differentiable in (to, +oo). There exists a neighbourhood N1 of t~ and a positive constant 7 such that, if the trajectory u verifies u(t) E N1 n bd K for some t > to, we have: d ~ ( u ( t ) ) + y~(u(t)) <~O.
Proof. The trajectory u is differentiable in (to, +o0) and u'(t) = F(u(t)) F(v) = f -
for all t > t0,
(7.4a)
(u(v),f)rl(v).
(7.4b)
This assertion follows from theorem 4 and corollary 1, together with the continuity of map F on bd K. Actually, F is differentiable on bd K, and a simple computation shows that the derivative of F at ti is given by
F'(a)
.z = -{W"(a) .z - (v(a),
W"(a)" z ) ~ ( a ) }
Vz.
(7.5)
Let us denote w(t) = u(t) - ~, and linearize the differential equation (7.4) in a neighbourhood of ti. From (7.5) we obtain
u'(t) = -{~0"(ti) • w(t) - (u(ff), ~/"i(a) • w(t))r/(t2)} +
Ilw(t)lle(w(t))
for all t > t0, Ile(z)ll tending to zero as Ilzll- Moreover the function t---~ ~£(u(t)) is differentiable at any t > to and one has
d ~ ( u ( t ) ) = (Paw(t), u'(t) ). It follows that
d • = (-(Paw(t), u ~p"(a)" ( tw(t)) )+ Ilw(t)llZe(w(t)), ) where e ( z ) ~ 0 as Ilzll. One checks that assumption (7.1b) admits a local version: (Pov, ~"(v). v)/> plllrll 2, for all boundary points v and all directions v in bd T(K', v), Pl being a positive constant, cf. [9]. Hence we deduce that
d .~( u( t) ) <~ - P l llw(t)ll z + IIw(t)ll2e( w( t) ), and we conclude the proof.
776
P. LABORDE
Proof of the theorem. According to the previous lemmas there exists a positive constant 6 so that the properties below hold: ~e(v) t>fill o - a l l ~ i f v E b d K
d~e(u(t)) <~-7~e(u(t))
i f / > t0
and
Ilv-alr~<6
(7.6)
and
Ilu(t) - alp ~< 6,
(7.7)
with/3 ~ (0, 1) and y > 0. Let us check that, if there exists t 1 i> t o such that [[u(tt) - all 2 ~ tl. The proof is by contradiction; so we suppose the existence of t 2 = inf{t t> tl :
Ilu(0
- all 2 > 6}.
On the one hand, since ~£(v) ~< IIo - all 2 for all v, the definition of tl yields: . ~ ( U ( t l ) ) ~
On the other hand, the continuity of trajectory deduce from property (7.6) that
u(t)
implies that [In(t2) - •112 = 6. Then, we
~ ( u ( t 2 ) ) I>/36.
Moreover we have leads to
Ilu(t)
-
alp ~< 6 for all t in the nonempty interval (tt, t2), so property (7.7) ~(u(t,))
>
~(u(t2)).
This strict inequality contradicts the two previous inequalities, which proves the assertion. Let us now suppose the existence of a time t 1 such that tl I-- to and Ilu(tl) - all 2 ~ / 3 6 . The above-mentioned assertion and property (7.7) imply that ~(u(t)) tends to zero when t---> +o0; the same holds true for Ilu(t) - all, due to property (7.6). REFERENCES 1. AUBIN J. P., Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions. Math. Analysis and Applications, Part A, Adv. Math. Suppl. Studies (Edited by L. NACnBIN), Vol. 7A, pp. 159-229, Academic Press, New York (1981). 2. Auan~ J. P. & CELLINAA., Differential Inclusions, Springer, Berlin (1983). 3. AUBIN J. P., CELLINAA. & NOHEL J., Monotone trajectories of multi-valued dynamical systems, Annali Mat. pura appl. 115, 99-117 (1977). 4. BRI~zls H., Operateurs Maximaux Monotones et Semi Groupes de Contractions dans les Espaces de Hilbert, NorthHolland, Amsterdam (1973). 5. GILBARGD. & TRUDINGERN. S., Elliptic Partial Differential Equations of Second Order, Springer, Berlin (1977). 6. HADDAD G., Monotone trajectories of differential inclusions and functional differential inclusions with memory, Israel J. Math. 39, 83-100 (1981). 7. HESTENES, M. R., Optimization Theory, Wiley, New York (1975). 8. LABORDEP., Etude d'un probl~me d'6volution non-monotone, Ann. Fac. Sci. Toulouse 5, 1-14 (1983). 9. LABORDEP., Analyse math6matique de probl/~mes en th6orie de la plasticit6, Thesis, Universit6 de Bordeaux I (1984). 10. LAaOmaE P., Analysis of the strain-stress relation in plasticity with non-associated laws, Int. J. Engng Sci (to appear). 11. MIGNOTF., Controle darts les in6quations variationnelles elliptiques, J. funct. Analysis 22, 130--185 (1976). 12. MOREAUJ. J., Evolution problem associated with a moving convex set in a Hilbert space, J. diff Eqns 26, 347374 (1977). 13. ROCKAFELLARR. Z., Convex Analysis, Princeton University Press (1970). 14. ZARANTONELLOE., Projections on convex sets, in Contributions to Non Linear Functional Analysis. Symposium, Madison, 1971, Academic Press, New York (1971).
0362-546X/87 $3.00 + .00 Pergamon Journals Ltd.
Printed in Great Britain.
A NONMONOTONE
DIFFERENTIAL
INCLUSION
P. LABORDE Laboratoire associ6 au C N R S No. 40226, Universit6 de Bordeaux I, 351, cours de la Lib6ration, 33405 Talence Cedex, France
(Received 25 November 1985; received for publication 11 April 1986) Key words and phrases: Differential inclusions, n o n m o n o t o n e set-valued map, discretization, uniqueness, asymptotic behavior. INTRODUCTION
LET E BE an Euclidean space, K a nonempty closed subset of E and ~p a convex function in E. Consider A the set-valued map with domain K vanishing in the interior of K and such that, for each boundary point v, Av is the set of outward normals to the level surface of ~p passing through v. We want to study the following differential inclusion in E, with state constraint du
d--t(t) + Au(t) ~ f(t),
u(t) E K,
(0.1)
where f is a given function in [0, T]. This problem arises in a preliminary study of models in theory of plasticity with "nonassociated laws", cf [10, 9]. We introduce a compatibility condition between K and ~p which is essential in this work. A special case, where this condition is satisfied, occurs when K is a convex set and 7Ythe gauge function of K. Hence Av is the normal cone to K at v, so A is maximal monotone, and (0.1) defines a sweeping process in the sense of Moreau [12]. In general, the operator A does not verify a monotonicity-type property. Available literature concerning nonmonotone differential inclusions essentially treats existence in a framework different from this article, cf., for example, [2]. We first prove the existence and uniqueness of a solution w to the inclusion
w+Aw~e,
wEK,
for a given element e in E. This result allows us to define an implicit discretization method for the differential inclusion. We show the convergence of this approximation, which yields the existence of a function u(t) verifying (0.1) a.e. in (0, T) and an initial condition. Under additional assumptions, we obtain a characterization of the velocity du/dt; moreover the uniqueness of solution u(t) is proved. Finally, we give a result concerning the asymptotic behavior of trajectories. The basic assumptions of this work lead to more technical proofs than in the monotone case. 1. T H E
SET-VALUED
MAP A
1.1. Definition and assumptions Let E be a real Euclidean space with the inner product denoted by (., • ) and the associated norm
I1"II757
758
P. LABORDE
Notation. Int X, bd X and cl X are respectively the interior, the boundary and the closure of a subset X of E, respectively. We associate with a convex cone X, its polar X ° -{~ E E : (~, v) ~< 0 Vv ~ X}. If X is a closed convex set, N ( X ; v) denotes the normal cone to X at a point v : N ( X ; v) = (X - v) °, gx the gauge function of X : gx(v) = inf{r > 0 : v E rX}, and proj(X; v) the (normal) projection on X of a point v. The subdifferential of a convex function q9 at a point v is denoted by Oqg(v), and inf q0 stands for the infimum of function q0 on its domain. With these notations, let K be a nonempty closed subset of E, and ~p a convex function in E such that int(dom ~p) D K. Definition. W e call A the set-valued map from E to E, with domain K such that: if v ~ int K : A v = {0}, if v E bd K : A v is the normal cone to the level set {w : ~p(w) ~< ~p(v)} of ~p through v. Remark. Let v be a boundary point of K such that ~p(v) > inf lp. Then, we have (cf. [13, p. 222]): A v = {X : )C = ~ with ). I> 0 and ~ E O~p(v)},
(1.1)
i.e. A v is the closed convex cone generated by O~p(v). Now, we introduce a compatibility condition between K and ~p which is essential in this work; (1.2c) is the main property. Basic assumptions. We first suppose that the minimum set of ~p is nonempty and included in K: O 4: {v : ~p(v) = inf ~p} C K.
(1.2a)
Moreover, if v E bd K, the cone v + A v (with vertex v) must be pointing strictly outward to K, in the following sense: there exists a constant o~ ~ (0, 1] such that: X E A v implies (X, w - v) ~< (1 - ~)IlXllIIw - vii
/
(1 .2b)
Vw ~ g .
Finally, if v, w ~ bd K, the cones v + A v and w + A w must "diverge"; more precisely: v, w E K, w - v q~ ( A v ) °" imply (p, ~)/> 0
V~ E A w .
(1.2c)
p = proj((Av)°; w - v) 1.2. Examples 1.2.1. Let us suppose that K is a closed convex subset of E, containing the origin in its interior, and ~p is the gauge function g r of K. Then, for each v, A v is the normal cone N(K; v) to K at v. In other words, A is the subdifferential of the indicator function of K. Assumptions (1.2) are obviously satisfied (with tr = 1 in (1.2b)) a n d A is maximal monotone. In the following, this will be referred to as the classical case.
A nonmonotone differential inclusion
759
1.2.2. Let ~p be the norm of E, and K a compact set containing the origin in its interior. So we have, for all boundary points v of K (cf (1.1)): A v = {x : X = ~ v , ~ >~O}.
Suppose that the gauge function of K is Lipschitz-continuous on the unity sphere of E. (In particular, this holds when K is convex.) Then assumptions (1.2) are fulfilled, cf. [8]. But A is not a Lipschitz perturbation of a maximal monotone set-valued map in general. 1.2.3. In the previous example, the function ~p is differentiable on the boundary of K. As a consequence, when v lies on bd K, A v is reduced to a half-line. We now consider a case where ~p is not differentiable. Let E be the space of symmetric endomorphisms of ~", equipped with its Euclidean structure. Denote sp(v) the set of eigenvalues of an endomorphism v, and set: 2,/~sp(v)
~p(v) = max{). - / ~ : 2,/~ ~ sp(v)}
Vv E E.
Then, the corresponding set-valued map A obeys conditions (1.2); cf. [9, 10]. 1.2.4. Finally, let us give two examples in E = R 2. In the first case the domain K of A is not convex:
g = {v = {Or, v2} ~ ~ :
02 - I o l l < 0, o21 + v~ < 2},
~p(v) = v~ + (v2 + 1) 2
Vv ~ 4 2.
In the second example, the minimal set of ~p intersects the boundary of K: K = {v E R2: vt <~ 0, vl + Iv21 ~<0},
~p(v) = max{Iv21, max{v l, 0}}. The basic assumptions are verified in each of these two cases. 2. F I R S T P R O P E R T I E S
We now deduce from the previous basic assumptions some useful properties of A. 2.1. Characterization o f A
Let cr be a continuous function from E to the interval [inf ~p, +oo) such that o(v) {~ ~p(v)
ifv E b d K
~p(v)
ifv~intK,
<~V(v)
ifv~K.
Such a function a always exists; we will give an example. Now, define for each v in E: Q(v) = {w: ~p(w) ~< or(v)}
which is a nonempty closed convex subset of E. The following characterization of A holds.
760
P. LABORDE
LEMMA 1. Under assumption (1.2a) the condition v E K is equivalent to v ~ Q(v); furthermore
A v = (Q(v) - v) °
for all r i n K .
The proof is easy. In example 1.2.1, where ~p is the gauge function of the convex set K, it is enough to take tr(o) = 1, i.e. Q(v) = K, for all v. In order to construct a function a into the general case, we associate with the set K a continuous function q~ such that {:1 tp(v)
ifv E b d K 1 ifvEintK,
(2.1) >1
ifvq!K.
One may take
qg(v) = 1 + e(v) dist(bd K; v), with e(o) = 1 if 0 $ K, - 1 if v E K, and where dist(X; w) denotes distance of a point w from a subset X in E. Naturally, when K is convex we may also choose tp = gK. Then, let:
a(v) = max{~(v) + c(1 - qg(v)), inf %0}
Vv ~ E,
with c > 0. According to (1.2.a), this function tr works.
2.2. A continuity property The previous characterization is a convenient tool to establish a result of closed graph in the functional space
{
L 2 ( O , T ; E ) = v:(O,T)--->E,
So
}
[Iv(t)ll2dt< + oo ,
equipped with its Hilbertian structure. THEOREM 1. Assume (1.2.a) and let u,, %, be two sequences in L2(0, T; E) such that:
(i)x,(t)~Au,(t) a.e. t E (0, T), Vn; (ii) u, --->u in L2(0, T; E), u being a continuous function; (iii) %, ~ %weakly in L2(0, T; E). Then, x(t) E Au(t) a.e. t E (0, T). (In particular, the graph of A is closed in E × E.) In the classical case, A = N ( K ; . ) is a maximal monotone operator, and the closeness property of the theorem is well-known, cf. [4].
Proof. Let Q be the set-valued map defined at v E L2(0, T; E) as (cf. Section 2.1) a ( o ) = {w : w E L2(0, T; E), w(t) ~ a ( o ( t ) ) a.e. t}. First, let us verify that Q is lower semi-continuous (l.s.c.) in L2(0, T; E). Precisely, suppose that a sequence u, converges to u in L2(0, T; E), and let v ~ L2(0, T; E) be verify v(t) E Q(u(t)) a.e.t.
A nonmonotone differential inclusion
761
The question is whether there exists a sequence on tending to v in L2(0, T; E) such that v,(t) ~ Q ( u , ( t ) ) a.e.t. But, let on(t) = p r o j ( Q ( u n ( t ) ) ; v(t)), a.e.t. The continuity of the function tr implies the continuity of the map{w, z}---> proj(Q(w); z) from E × E to E. Hence, owing to the dominated convergence theorem, the sequence vn converges in L2(0, T; E) to proj(Q(u); v) which is equal to v. Thus Q is l.s.c. Let us make use of conditions listed in theorem 1. The 1.s.c. of ~), and the characterization of A (lemma 1) applied to Zn U A u , , yield: oT (g(t), v(t) - u(t) ) dt <<-O.
We deduce the pointwise inequality: (Z(O, v(t) - u(t)) <- 0
a.e. t,
Vv ~ Q(u(O),
due to the continuity assumption on u and a standard theorem of integration theory. The conclusion of the theorem follows from the characterization of A. 2.3. Pseudo-distance associated with A Let us denote, for all ul and u2 in K:
p~ = proj((Aui)°; uj - ui)(] ~ i)
(2.2a)
dA(ul ;u2) = min{llp~ II, ILOzlI}.
(2.2b)
In the classical case, we find: d,a(ul, u2) = Ilul - u211 for all ul, u: ~ g.
In general, the quantity dA(Ul, u2) gives an estimate of the distance between two elements ul and u2 of K in the following way. Prtoposrnon 1. Under assumption (1.2) we have the two inequalities
/211u - u211 d a ( u l , u2)
Ilu - u211
Vua, u2
K,
where a, is the constant of assumption (1.2.b). Proof. T w o cases may happen: either
(a)
u2 - ul E (Aux) ° and ul - u2 E ( A u : ) °,
or (with an eventual permutation of indices)
(b)
u2 - ul
(AuO °.
In case (a), we have dA(ul, u2) = I}ul -- u21}, and the two inequalities hold. Examine the second case. First, check the implication below: u2 - ul q~ ( A u l ) ° ~ u l - u2 E (Au:) °.
(2.3)
762
P. LABORDE
Precisely, by definition of A, the condition (u2 - ul) qi_ (Aul) ° implies that u I ~ bd K and ~P(ul) < lP(u2), whence we deduce that (Ux - u2) ~ (Au2) °. It follows that Ilul - uzll z --Iho ll z.
(2.4)
Now, recall the result of orthogonal decomposition according to mutually polar cones. Let C be a nonempty closed convex cone in E and w ~ E; then (cf. [7]). ~ w = p + q with p = proj(C; w) ¢:> t{p, q} ~ C x C O and (p, q) = 0.
(2.5)
By decomposition of u2 - Ul according to A u l and (Aul) °, assumption (1.2b) yields to ]lu2 - uall 2 1> [Lolll2 ~ og[lu1 - - U2[)2.
(2.6)
The proposition is a consequence of (2.4) and (2.6). 3. SOLUTION OF SOME TIME-INDEPENDENT INCLUSION In this section we study the solution w of the inclusion w + Aw ~ e
(w E K)
(3.1)
for an element e given in E. We shall also give a tangential condition satisfied by A and connected with the study of inclusion (3.1). 3.1. Projection associated with A THEOREM 2. Suppose (1.2) holds and let e in E. There exists a unique u lying in K solution of inclusion (3.1). If e I and ez are any two elements of E, let ui be the solution of ui + Aui ~ e~ (i = 1, 2); then IlUl - u2112
-111el - e2LI
where c~ is the constant of assumption (1.2b). Definition. The projection associated with the set-valued map A is the map PA : e---~ u such that u + A u ~ e. Owing to theorem 2, PA is Lipschitz-continuous in E with the Lipschitz constant L -- c~-1/2. This definition is motivated by lemma 2 hereafter and also by the following remark. Remark. The map PA corresponding to the classical case (Section 1.2.1) is the normal projection on the convex set K. The Lipschitz constant of PA given by the theorem is L -- 1 in this case; that is the classical contraction property of proj(K; • ). Proof of the theorem. If e E K, a trivial solution of inclusion (3.1) is u = e. Suppose that e K and associate with K a continuous function q0 : E--~ R verifying (2.1). Let w(r) be the projection of e on the level set {z : , ( z ) ~< r}. According to assumption (1.2a), the minimizer w(inf ~p) of function ap lies in K; thus cp(w(inf ~p)) ~< 1. Moreover, since e 4= K, we also have qg(w(~V(e)) > 1. A continuity argument ensures the existence of f ~ [inf ~p, ~p(e)) such that q0(w(r--)) -- 1, i.e.
A n o n m o n o t o n e differential inclusion
763
W(r-) is a boundary point of K. Then, the definition of w(r) implies that (e - w(r-)) E Aw(r-), i.e. u = w(r-) is a solution to inclusion (3.1). Now, let el, e2 E E and ui be a solution of ui + Aui ~ ei (i = 1, 2). We use the decomposition in cases (a) and (b) of the proof of proposition 1. In case (a), one has: ( ( e I -- U l ) -- ( e 2 -- u 2 ) , u 1 -- u 2 ) ~ 0 , SO
IlUl - u ll
lie1 - e211-
Let us consider case (b). By hypothesis (1.2c) the elementpt = proj((AUl)°; u2 - ul) verifies: (p,, (e2 - u2) - (el - u l ) ) / > 0. Then Ilpl]l2 ( = (p,, u2 - u,)) is less than (Pl, e2 - el), i.e. ]IP,II ~< lie1 -- e21], so (2.6) implies
Ilu, - u2ll
~
ot-1/Zlle~
-
e21l.
(3.2)
This inequality is also true in case (a). In this way we deduce the uniqueness of the solution u to u + A u ~ e and the Lipschitz property of the theorem. Moreover, we have LEMMA
2. The map PA verifies properties (i), (ii), (iii):
(i) p2 = PA, PA(E) = K; (ii) v E K ~ P A V = V ; V ~ K ~ P A V E (iii) X E A(PAV) ~ PA(V + X) = PAY;
bdK;
in particular:
v E K ~ P A ( V + A v ) = {v}. The proof of the lemma rests with the following facts (cf. [9]). First, A v contains the origin for all v in K. Second, A vanishes inside the interior of K. Finally, the values of A on the boundary of K are convex cones. 3.2. Boundary condition Let us denote by T(X; v) the contingent cone to a closed subset X of E at a point v. By definition, T(X; v) is the set of directions 0 such that there exist two sequences 0n and hn verifying (cf. [1]):
On--'>OinE, hn-'-~O, h n > O v + h n On ~ X
Vn,
Vn.
When X is convex, T(X; v) reduces to the tangent cone to K at v. We indicate a first geometrical property of A. LEMMA 3. Assumptions (1.2) o n A imply
(g - Ao) n T(K; v) 4 : 0
VvEK,
VgEE.
(3.3)
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P. LABORDE
Let us comment on this result. For a given e in E, put
F(v) = e - t) - At)
Vv E K.
A solution u of the inclusion u + A u ~ e is an equilibrium state for the set-valued map F, i.e. u verifies: F(u) ~ O. On the other hand, property (3.3) yields
F(v) n T(K; v) q~ 0
Vv E K.
(3.4)
This expression is the tangential condition which appears in results for equilibria of setvalued maps, cf. [2]. Nevertheless, the assumptions of continuity and compactness used in these references are not verified in our problem.
Proof of lemma 3. Let Uh = V + hg and Oh = P AUh. The Lipschitz-continuity of P A implies that the s e t { h - l ( t ) h -- O)}h>0 is bounded in E. Hence, we can extract a sequence converging to an element 0 in E. The condition oh E K ensures that 0 E T(K; v). Moreover: g - h - l ( V h - t))(= h - l ( U h -- Oh) ) E A t ) h . Since the graph of A is closed in E x E (cf. theorem 1), the convergence of Oh to V shows that: g - 0 E At). The lemma follows. Now, let us give a boundary condition simpler than (3.3) which will be used further. LEMMA 4. The condition:
( - A v ) O int T(K; v) ~ 0
Vv E K
(3.5)
implies (3.3). The converse is true when K is a convex set with a nonempty interior (in particular, assumption (1.2) and the convexity of K imply (3.5)).
Proof. First, suppose that (3.5) holds: there exists 0 E ( - A v ) O int T(K; v). Let g in E. The condition 0 E int T(K; v) implies the existence ofA > 0 such that (g + A0) E T(K; v). On the other hand, since 0 lies in cone ( - A v ) , one also has (g + ~,0) E g - Av. In this way we obtain (3.3). Conversely, suppose condition (3.3) and the convexity of K. The interior of K is nonempty, so there exists 0 such that T(K; v) + 0 C int T(K; v). Choosing g = - 0 in (3.3), we also have ( - A v ) n (T(K; v) + 0) g 0 . Property (3.5) follows. 4. A P P R O X I M A T I O N OF THE D I F F E R E N T I A L INCLUSION
We are now in a position to find a solution u(t) to the differential inclusion:
u'(t) + Au(t) ~ f(t)
(u(t) E K),
(4.1)
with the initial condition: u(0) = u0; the prime in (4.1) stands for the derivation with respect to t.
(4.2)
A nonmonotone differential inclusion
765
Let us put conditions on data: f ~ L 2 ( 0 , T;E)
and
Uo ~ K .
(4.3)
4.1. Discretization Let k > 0 and N be the integer such that ( N - 1)k < T ~ Nk. Denote tn = nk for n = 0, 1. . . . , N - 1, tN = T and 1~ = (tn- 1, tn). Let also fk be a sequence of functions converging to f in L2(0, T; E), such that fk(t) = f ~ constant on In. The discretized problem is to find a continuous piecewise affine function Uk in [0, T] solution of (4.4) (4.5). U k ( t ) = k - l ( t ~ -- t)u~,-1 + k - l ( t - t n _ l ) u ~ k - l ( u ~ - u~ -1) + AuT, ~ fT,
Vt~I n
(u~ v= K)
(4.4a) (4.4b)
for n = 1 , . . . , N and u ° = u0.
(4.5)
With the definition introduced in Section 3.1, the implicit algorithm (4.4b) may be written: u~ = PA(u~ -1 + kf"k).
(4.6)
Remark. In the particular case A = N ( K; . ), the differential inclusion (4.1) describes a sweeping process by a mooing convex set, in the terminology of Moreau. Moreover, expression (4.6) defines the associated catching up algorithm of which the convergence is studied by this author in [12].
THEOREM 3. Suppose that hypotheses (1.2) and (4.3) are satisfied. Then, for any fixed k > 0, the discretized problem (4.4) (4.5) admits a unique solution uk. Furthermore, there exists a subsequence k(n) converging to zero and an absolutely continuous function u in [0, T], such that: Uk(,) ~ U uniformly, in [0, T], u~,(n)~ u'
weaklyin L2(O, T; E).
The limit u verifies the differential inclusion (4.1) a.e. in (0, T), and the initial condition (4.2). Finally, we have the estimate:
Ilu'(t)ll
zllf(Oll
a.e. t,
where L is the Lipschitz constant of PA. Remark. In [10] (cf. also [9]) is studied an approximation method of the differential inclusion by penalization-regularization. Note also that [3, 6] give others' existence results for differential inclusions in the same direction. Proof. The Lipschitz continuity of PA (theorem 2) and equality (4.6) imply:
IIk-l(u~ - u~-l)ll <~Zllf~ll
n = 1, 2 . . . . , N.
(4.7)
Since the sequence fk is bounded in L2(0, T; E), the same holds true with respect to the
766
P. LABORDE
sequence u~,. Thus, one can extract a subsequence Uk(,) which converges uniformly in [0, T] to some absolutely continuous function u such that: u~,(,) ~ u' weakly in L2(0, T; E).
(4.8)
Theorem 1 allows us to pass to the limit in the discretized equation (4.4b). Therefore the function u verifies the differential equation (4.1.) a.e. in (0, T). Finally, we deduce from (4.7), (4.8) that [lu'(t)l[ ~< LIIf(t)ll a . e . t . 5. E X P R E S S I O N
OF THE VELOCITY
The purpose of this section is to express the derivative, at a time t, of a solution u to differential inclusion (4.1) in terms of the position u(t) and of the right-hand side f(t). Henceforth, we suppose in addition that K is a (closed) convex set.
(5.1)
Notations. For a function g, put g(t + O) =
lim
h--*0,h>0
h -~
i
t+h
g(s) ds,
when this limit exists, t is called a Lebesgue point from the right of g. Furthermore, we denote g' (t + 0) the derivative from the right of g at t, q0'(v; d) the directional derivative of a function q9 at a point o into a direction d, q9'(v) the gradient of function 99 at v, when these quantities exist. Finally, x + stands for max{x, 0}. Let us first study the differentiability properties of P3 at v such that v verifies either (a) or (b);
]
(a)
v ~ int K,
}
(b)
v E bd K and ~p is Gateaux-differentiable at point o.
(5.2)
Notice that case (5.2a) is trivial, since PA is equal to the identity on int K. If o verifies (5.2b) the gradient ~p'(v) is different from zero (otherwise A o = E, which contradicts assumption (1.2b); moreover A o -- ~+~p'(v) (cf. (1.1)), and the boundary condition of lemma 4 becomes -~p'(o) ~ int T(K; o).
(5.3)
PROPOSITION 2. Let o be such that (5.2) holds and d E E. There exists a unique element 0 which satisfies the equivalent conditions (i) and (ii): (i) 0 = d - / . ~ ' ( v ) with 1. = inf{tt >1 0 : d - # ~ ' ( v ) ~ T(K; v)} (ii)
d = 0 + • with 0 E T(K; v), X, E A v , if d C T(K; v) : X = O, if not : 0 ~ bd T(K; v).
Moreover, PA admits a directional derivative at v into direction d equal to P'A(O; d) = O.
A nonmonotone differential inclusion
767
Proof. (a) Let us first verify that conditions (i) and (ii) define the same unique element 0. It is obvious in the case d E T(K; v) where 0 = d. Examine the opposite case: v E bd K and d q?. T(K; v). Let v(/0 = d - / ~ p ' ( v ) and A = {/u/> 0 : v(/~) ~ T(K; v)}. The boundary condition of lemma 3 ensures that A 4= O. Let ~, = infA; by definition, v(2,) is the (unique) element 0 which verifies condition (i). In fact v(~,) ~ T(K; v), due to the closeness of T(K; v); moreover ~, > 0, since v(0) $ T(K; v). The definition of ~, implies that v(/~) $ T(K; v) for all/~ @ [0, ).[, it follows, by continuity: v(;t) E bd T(K; v). Then, condition (ii) is satisfied by 0 = v(A), Finally, consider any element 0 verifying (ii). There exist v/> 0 such that 0 -- v(v). The convex cone T(K; v) has v(v) on its boundary and ( - ~ ' ( v ) ) in its interior. Consequently v > 0 and we have for all/t E [0, v[: r(/~)(= v(v) + (9 - ~)~p'(v)) ~ T(K; v). We conclude that t, = ~., i.e. v(~.) is the unique element 0 which obeys condition (ii). (b) We now study the differentiability of PA at a point v verifying (5.2) into a direction d in E. Denote Wh = PA(V +hd). First, we suppose: d ~ int T(K; v). The Lipschitz-continuity of PA implies that
IIh-l(wh
-
v)lt < Zlldll
Vh > 0.
Then, there exists a subsequence h(n) ~ 0 and v E E such that h(n)-l(Wh(n) -- v) ~ "r. But we have, by definition of Wh:
h-l(Wh -- V) ÷ A w h ~ d. Owing to the closeness of the graph of A (cf. theorem 1), it follows that v + A v 3 d. Furthermore v E bd T(K; v), since d ~ int T(K; v), i.e. w h ~ bd K for all h > 0 (lemma 2, (ii)). Then the limit v is equal to the element 0 given by condition (ii) of the proposition; and the whole family h-l(Wh -- l)) converges to 0. The same holds true when d E int(K; v), because we have for all h positive and small enough: v + hd E K, so Wh = V + h d (lemma 2, (ii)), i.e. h-l(Wh -- I)) = d. In conclusion, for any d in E, the directional derivative P'A(V; d) exists and is equal to the element 0 above defined. That ends the demonstration. We easily deduce from this proposition that: COROLLARY 1. Let v be a boundary point of K such that ~p and the gauge function gr are Gateaux-differentiable at v; denote: and
v(v) = kg'K(V),
where k is positive and such that (v(v), r/(v)) = 1. Then, there exist directional derivatives of PA at v and
(P'A)(v; d) = d - (d, v(v))+r/(v)
foralldinE.
If v is an interior point of K, we have:
(PA)'(V; d) = d
for alld.
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P. LABORDE
The study of the differentiability of Pa enables us to express the velocity of the solution u(t) of differential inclusion (4.1). THEOREM 4. Assume (1.2) and (5.1) hold. Let u be an absolutely continuous function in [0, T] solution of: u' + A u ~ f a.e. in (0, T) (theorem 3). Let also to E [0, T) be a Lebesgue point from the right of f, such that v = U(to) verifies condition (5.2). Then, the function u has a derivative from the right at time to given by
u'(to + O) = (PA)'(U(to);f(to + 0)). Remark. This theorem and the previous proposition are to be connected with the following results known in the classical case. First, the normal projection projr on the closed convex set K possesses directional derivatives at any point v E K and (cf. [14, 11]): ( p r o j r ) ' ( v ; d) = proj(T(K; v); d)
Vd E E.
Furthermore, by application of a theorem of Brrzis (cf. [4]) to the case A -- N(K; • ), one has
u'(to + 0) = proj(T(K; U(to));f(to + 0)), for any Lebesgue point from the right to ~ [0, T) of f. To prove theorem we shall need a preliminary property. LEMMA 5. Let h(n) be a sequence of strictly positive numbers tending to zero such that U(to + h(n)) ~ bd K for all n. Then, there exists lim h(n)-l[U(to + h(n)) - u(t0) ] = (P'a)'(U(to);f(to + 0)). n
Proof. Let us denote, for all h G (0, T - to): Oh =
h-l[U(to + h) - U(t0)],fh = h -1
f
to+h
f(t) dt,
to
Xh = h-1
f
to+h
X(t) dt
where X = f -
u' a.e.
t0
Moreover, if X is a subset of E and/3 E (0, 1), denote by Xt~ the cone X¢ = {~ E E : there exists g E X s u c h that (X, ~) I> (1 -/3)Ilxll I1~11}. With these notations, the mean-value equation derived from (4.1) is written Oh ÷ Xh ----fh"
(5.4)
On the other hand, the solution u(t) is a continuous function and X(t) E Au(t) a.e.t. From the upper-semi-continuity of subdifferential 0~p (cf. [13]) we deduce, for all/3 > 0, the existence of ho(/3) > 0 such that
g(t) E (Zu(to) ) ~ a.e. t E (to, to + ho(fl) ).
A nonmonotone differentialinclusion
769
The convexity of (Au(to))~ and the mean-value theorem yield:
Xh E (Au(to))13
Vh E (0, ho(fl)) (Vfl > 0).
(5.5)
Now, let h(n) be a sequence tending to zero, such that U(to + h(n)) E bd K for all n. In a first step we suppose that U(to + h(n)) 4: U(to) for all n. Owing to condition (5.3), there exists fl > 0 such that
-(Au(to))31j C T(K; U(to) ). The sequence of boundary points U(to + h(n)) tending to u(t0), it follows that
Vh(,) q~ -(Au(to))2#
Vn >I nl,
(5.6)
for some integer nt. On the other hand, according to (5.5), there exists another integer n2 such that:
Xh(,) E (Au(to))t3
Vn >~n2.
(5.7)
This last property, together with (5.6) and the mean-value equation (5.4), implies the existence of a positive constant 7 such that:
[Ivh(.)l[ +
Ilxh(.)ll fllA(.)ll
Vn 1> max{hi, n2}.
One deduces that sequences Oh(,) and ~Ph(,) are bounded, and there exist subsequences such that
Vh(,') ~ Vo,
gh(,') ~ gO.
We derive from (5.4) that v0 + X0 = f(to + 0). The hypothesis U(to + h(n)) E bd K for all n leads to v0 E bd T(K; U(to)). Passing to the limit as fl---~ 0 in (5.7), we also obtain X0 E Au(to). It follows that v0 = Pa(u(to);f(to + 0)) by proposition 2 and the whole sequence Vh(,) tends to v0. In a second step, let us study the case where U(to + h(n')) = U(to) for a subsequence h(n') of h(n). Sincefh tends tof(t0 + 0) as h ~ 0, expressions (5.4) (5.5) imply thatf(t0 + 0) E Au(to), so (Pa)'(U(to);f(to + 0)) = 0 due to proposition 2. Using the result of the first step, we deduce that the whole sequence vh(,) tends to 0. The proof is complete.
Proof of the theorem. We keep the notations of the previous lemma. First, consider the case where: there exists h0 > 0 such that: u(t) ~ int K
Vt E (to, to + h0).
(5.8)
Since Z(t) = 0 a.e. t ~ (to, to + h0), equation (5.4) becomes: Vh =fh for all h in (0, h0). Then f(to + O) E T(K; U(to)), and there exists u'(to + 0) equal to f(to + 0). By proposition 2, we obtain the conclusion of theorem when (5.8) holds. Let us now examine the opposite case. We consider any sequence of numbers k(n) which converges to zero, and we define
h(n) = sup{h E [0, k(n)] : U(to + h) ~ bd K}.
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P. LABORDE
The negation of (5.8) ensures that h(n) > 0. Since u(t) E int K for all t lying in the interval (to + h(n), to + k(n)), we have X(t) = 0 a.e. t in this interval. By definition Of%h, it follows that
h(n)
Xk(,) = k - ~ %h(,).
(5.9)
As U(to + h(n)) E bd K for all n, we can use the previous lemma for sequence h(n). Then, the sequence %h(,) tends to Z0aP'(u(t0)) with (proposition 2): ~.0 = i n f ~ / > 0 :f(t0 + 0) - #~p'(U(to)) E T(K; u(t0))}. So is the sequence Zk(,), due to (5.9). Consequently ok(,) converges to (Pa)'(u(to);f(to + 0)); this ends the proof. 6. A UNIQUENESS RESULT The uniqueness of the solution of the differential inclusion is now established under some
regularity condition on the multi-valued map A; we also suppose that the right-hand side f of the differential inclusion is piecewise constant. In the classical case, the uniqueness results from the well-known property of decrease of the distance between two trajectories with respect to time, due to the monotonicity of A -- N(K; .). In our problem, we show that the pseudodistance da (cf. Section 2.3) between two possible solutions satisfies a Gronwall condition. Assume that: ~p is a %2-function in bd K
(6.1)
{v : ~p(v) = inf ~p} C int K.
(6.2)
THEOREM 5. Suppose (1.2), (5.1), (6.1), (6.2) and let f E E, u0 E K. There exists a unique Lipschitz-continuous function u in [u, + ~ ) such that:
u'(t) + Au(t) ~ f
a.e. t > 0; u(0) = u0.
(6.3)
A straightforward consequence is: COROLLARY 2. If f is a piecewise constant function in [0, T], the solution of u' + Au ~ f a.e. in (0, T), and u(0) = u0, .o unique. The Lipschitz-continuity of a solution, in theorem 5, is deduced from theorem 3. The uniqueness result is based on two lemmas hereafter. First, trajectories for the differential inclusion with a constant right-hand side behave as it follows. LEMMA 6. Let u be a Lipschitz-continuous function in [0, + ~ ) satisfying: u'(t) + A u(t) ~ f a.e. t > 0. These exists a time to E [0, +oo] such that u(t) E int K when t ~ (0, to), and u(t) E bd K when t/> to.
Proof. We eliminate the trivial case: u(t) E int K for all t > 0, which corresponds to to -- +oo in terms of the lemma. Consider the opposite case and define to --- inf{t > 0 : u(t) E bd K}.
A nonmonotone differential inclusion
771
The question is whether the trajectory u(t) lies on the boundary of K for all t I> to. We proceed by contradiction. So, suppose the existence of a time t2 > to such that u(t2) E int K, and let t 1 = sup{t <~ t 2 : u(t) E bd K}. Necessarily we have tl E [to, t2); moreover t~ > 0 since tl = to implies to > 0. Let us consider the differential inclusion (6.3) and set
X(t)=f-u'(t)
a.e. t > 0.
By an integration it follows, for all t3 1> 0 (t3 4: tt):
(
~"
u(t3) t3
u(tl).~ + _ _ tl
/
t3
(u, X(t)) dt = ( u , f ) ,
(6.4)
tl
where u is a given nonzero vector normal to K at U(tl). On the one hand, take in this equality t 3 = t 2. Since g = 0 a.e. in (q, t2), and u(t2) ~ int K, we obtain the strict inequality ( v , f ) < O. On the other hand, the boundary condition (3.5) at the point v = U(tl) and the continuity of derivative ~p' at this same point, prove the existence of some time t 3 E [0, tl) such that (v, X) ~> 0 a.e. in (t3, tl). Equality (6.4), for this choice of t3, yields: 0 ' , f ) ~> 0. This implies a contradiction, and the lemma holds. Now, the regularity assumptions (6.1), (6.2) enables us to prove that: LEMMA 7. Let u~ and/-/2 be two Lipschitz continuous functions in [0, + ~ ) such that (i = 1, 2)
u[ + Aui ~ f
a.e. in (0, +oo),
ui(O) = Uo.
Let also
r(t) ----dA(Ul(t),
u2(t)) 2
(cf. (2.2)). Then, the function r is locally Lipschitz-continuous in [0, +oo). Moreover, for any T < +0% there exists a constant C such that:
r'(t) <~ Cr(t)
a.e. t E (0, T).
Proof. (a) Taking into account the equality u[ = f a.e. in {t > 0: ui(t) E int K}, we deduce from lemma 6 the existence of to E [0, +oo] such that: ul(t) = u2(t ) ul(t)
and
V t ~ [0, to],
u2(t) E b d K
Vt>1 to.
Whenever to = +oo we have Ul(t ) = u2(t ) for all t/> 0, and the lemma is obviously verified. So suppose that to < + ~ . By definition, we have r = min{llPlll 2, liP2112}with (j 4= i)
pi=(uj--ui)-(uj-ui,
Tli)+~li; ?li=][~tY(ui)ll-llp'(ui)
t<~to .
772
P. LABORDE
Let T E (to, +oo); the function ~i is Lipschitz-continuous in [to, T], due to the regularity of trajectory ui and hypotheses (6.1), (6.2). Precisely:
I1~;11 ~ c, (T)
a.e. in(to, T),
(6.5)
where C(T) is a constant defined by the Lipschitz constant of ui(t) and by M and ro such that M =
sup{llw"(u~(/))ll,
t ~ [to, T]} < + ~ ,
ro = inf{llw'(ug(t))[I, t ~ I/o, T]} > 0. Thus the function IIp,II2 is locally Lipschitz-continuous in [to, +~), and the same holds true with regard to r (cf. [5]). Since d (pi,~i)-~t(u.i-ui,~i)+=O
a.e. t ~> to,
the derivative of r is d
2
a.e. in {t/> to : Ilpi(t)ll ~ Ilpj(t)ll(J * i)}
(6.6a)
ld 2 dt Ilpill2 = (pi, u] - u~) - (rli, uj - u i ) + ( p i , rl[ ).
(6.6b)
r' = ~ IlPill with
(b) Let us prove that function r verifies the Gronwall's condition. Denote (i = 1, 2) Ji = {t >I to : uj(t) - ui(t) ~. ( A u i ( t ) ) ° Io = Jt f'l J2,
(j ~ i)},
Ii = [to, +oo)\Ji.
The set Ji is closed, by continuity of functions Uk(t) and ~p'(v). Moreover, we deduce from (2.3) that J1 O J2 = [to, +oo); that is the decomposition I0, 11, 12 defines a partition of [to, +oo). In Io, we have: Ilplll2 = lip2112= Ilul - uzl]2. It follows from (6.6) that: d r'=~llux-u2112~-(aul-Au2,ul-u2)
a.e. in/0,
hence r' ~ 0 a.e. in Io. In Ix, where r = Ilplll2, we have simultaneously:
llp111~ ~> ~lu~
-
u~ll ~
(cf. (2.6)), and (Pl,U~-U[)E-(Pl,Au2-Aua)<~O
a.e. in11
(due to assumption (1.2c)). For a fixed T in [to, +oo), these facts together with (6.5) and (6.6) imply the existence of a constant C such that d
r' = ~ IIp,II 2 -< Cl[p,II ~ -- Cr
a.e. in I1 t~ (0, T).
A nonmonotone differential inclusion
773
The situation is symmetrical in 12, and the lemma follows.
Proof of the theorem. The previous lemma shows that the function r verifies the Gronwall's condition. So, proposition 1 implies that Ul = u2. 7. A S Y M P T O T I C
BEHAVIOR
7.1. Equilibrium Let f b e a nonzero vector. An equilibrium ~ for the differential inclusion u'(t) + Au(t) ~ f is a solution of the inclusion A l / D f (t/E K). We strengthen assumption (1.2c) by condition (7.1): ~p is strongly convex (i.e. there exists Y > 0 such that: ~0(Zu + (1 - ;0v) ~< Z~p(u) + (1 - ).)~p(v) - Z(). - 1)~[u -
vii2/
(7.1a)
for all u, v in E and 1. in [0, 1]); there exists a positive constant p such that we have:
w - v q~ (Ao) °
}
I(P' X) >~Pllw - oll Ilpll Ilxll
/ (7.1b)
p = proj((mv)°; w - v) - implies~for allz ~ aw. Moreover, assume that K is a compact set.
(7.2)
In the classical case, an equilibrium r/is a contact point of K with the supporting hyperplan to K normal to f, and assumptions (7.1), (7.2) are equivalent to the strong monotonicity of 0gr. In the general case we now prove the following existence result. THEOREM 6. Under hypotheses (1.2a, b), (7.1), (7.2), there exists a unique equilibrium ti.
Proof. (a) Existence. We use a procedure similar to that in the proof of theorem 2. Let r ~> inf ~p and w(r) which maximizes v ~ (/~, o) subject to lp(v)~< r. Owing to the strong convexity of ~p, we define a unique element w(r), and the map r--* w(r) is continuous in [inf ~p, +~). The compactness of K implies that r0 = sup{~O(v) : v E K} < +~. Let tp be a continuous function associated with K by condition (2.1). It follows that q~(w(inf ~p)) <~ 1 ~< q~(w(ro)). There exists ? such that tp(w(?)) = 1, by continuity; so we have: Aw(f) ~f. (b) Uniqueness. Let ut and u2 be two equilibriums. We again use the decomposition into cases (a) and (b) of the proof of proposition 1. In case (a), the equality ut --- u2 is provided by the strong convexity of ~p. Let us consider case (b) and define Pl = proj((Aux)°; u2
- Ul).
774
P. LABORDE
On the one hand, the condition )~E Au2, together with assumption (7.1b) and property (2.6), implies that ( P l , f ) ~> plllUl - u2112,
p , > 0.
On the other hand, since ]~E AUl, we also have ( P l , f ) ~< 0. It results that Ul = u2.
7.2. Behavior of trajectories Let us consider a trajectory u(t), that is a Lipschitz-continuous function such that u'(t)+Au(t)~f
a.e.t>0.
We introduce a map ~ which will play the role of a Liapunov function for this problem. Denote Pv the (normal) projection on the hyperplan orthogonal to the gradient ~p'(v), i.e. Poz
( n ( v ) = IIw'(v)ll-X~,'(v)).
= z -
(z,
~(v))~(v)
T h e n w e define ~e as ~e(v) -- ½11e~(v - a)ll z
for all v ,
where a is the equilibrium, cf. Section 7.1. We require further regularity conditions, first on ~p with assumption (6.1), and next on K: the gauge function q9 of K is convex and twice Frechet-differentiable.
(7.3)
THEOREM 7. Assume (1.2a, b), (6.1), (6.2). (7.1), (7.2), (7.3) hold. Let a be an equilibrium and u(t) a trajectory. There exists a neighbourhood N of a so that, if U(tl) lies in N for some tl >t 0, we have
~(u(t)) ~ O,
u(t) ~ a
when t----~ +~.
Remark. Aubin [1] introduces a concept of U-monotone maps which enables him to construct Liapunov functions associated with set-valued maps. This author derives some results of asymptotic behavior for differential inclusions. The notion of U-monotome map is not suitable for our purpose and this is why condition (7.1) is introduced. We need two preliminary results in order to prove theorem 7. LEMMA 8. The map ~ is convex and Frachet-differentiable in E. Moreover there exists a neighbourhood No of a and a constant fl in (0, 1) such that
~e(v) I>/~llv -
all 2 for all v ~ (bd K) M No.
Proof. The convexity and the differentiability of ~ come out from the linearity of Pa. The differentiability of the gauge function q9 implies (q~'(a), v - a) = Ilv - all
e(v - a)
when v lies in bd K, e(0) tending to zero together with II011. Using the notations of corollary 1, we obtain: ( 0 ( a ) , v - a ) z ~ IIv - a l l 2 ( a + e ( o - a ) )
A nonmonotone differentialinclusion
775
for all boundary points v. In this relation the constant 6, namely 1 - (r/(ti), v(ti)) 2, is strictly less than 1 by assumption (1.2b). The lemma follows. Let to >/0 be the time such that u(t) E int K when t E (0, to) and u(t) E bd K when t t> to (cf. lemma 6). The compactness of K ensures that to is finite when f:/= 0. LEMMA 9. The function t---->~£(u(t)) is differentiable in (to, +oo). There exists a neighbourhood N1 of t~ and a positive constant 7 such that, if the trajectory u verifies u(t) E N1 n bd K for some t > to, we have: d ~ ( u ( t ) ) + y~(u(t)) <~O.
Proof. The trajectory u is differentiable in (to, +o0) and u'(t) = F(u(t)) F(v) = f -
for all t > t0,
(7.4a)
(u(v),f)rl(v).
(7.4b)
This assertion follows from theorem 4 and corollary 1, together with the continuity of map F on bd K. Actually, F is differentiable on bd K, and a simple computation shows that the derivative of F at ti is given by
F'(a)
.z = -{W"(a) .z - (v(a),
W"(a)" z ) ~ ( a ) }
Vz.
(7.5)
Let us denote w(t) = u(t) - ~, and linearize the differential equation (7.4) in a neighbourhood of ti. From (7.5) we obtain
u'(t) = -{~0"(ti) • w(t) - (u(ff), ~/"i(a) • w(t))r/(t2)} +
Ilw(t)lle(w(t))
for all t > t0, Ile(z)ll tending to zero as Ilzll- Moreover the function t---~ ~£(u(t)) is differentiable at any t > to and one has
d ~ ( u ( t ) ) = (Paw(t), u'(t) ). It follows that
d • = (-(Paw(t), u ~p"(a)" ( tw(t)) )+ Ilw(t)llZe(w(t)), ) where e ( z ) ~ 0 as Ilzll. One checks that assumption (7.1b) admits a local version: (Pov, ~"(v). v)/> plllrll 2, for all boundary points v and all directions v in bd T(K', v), Pl being a positive constant, cf. [9]. Hence we deduce that
d .~( u( t) ) <~ - P l llw(t)ll z + IIw(t)ll2e( w( t) ), and we conclude the proof.
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P. LABORDE
Proof of the theorem. According to the previous lemmas there exists a positive constant 6 so that the properties below hold: ~e(v) t>fill o - a l l ~ i f v E b d K
d~e(u(t)) <~-7~e(u(t))
i f / > t0
and
Ilv-alr~<6
(7.6)
and
Ilu(t) - alp ~< 6,
(7.7)
with/3 ~ (0, 1) and y > 0. Let us check that, if there exists t 1 i> t o such that [[u(tt) - all 2 ~ tl. The proof is by contradiction; so we suppose the existence of t 2 = inf{t t> tl :
Ilu(0
- all 2 > 6}.
On the one hand, since ~£(v) ~< IIo - all 2 for all v, the definition of tl yields: . ~ ( U ( t l ) ) ~
On the other hand, the continuity of trajectory deduce from property (7.6) that
u(t)
implies that [In(t2) - •112 = 6. Then, we
~ ( u ( t 2 ) ) I>/36.
Moreover we have leads to
Ilu(t)
-
alp ~< 6 for all t in the nonempty interval (tt, t2), so property (7.7) ~(u(t,))
>
~(u(t2)).
This strict inequality contradicts the two previous inequalities, which proves the assertion. Let us now suppose the existence of a time t 1 such that tl I-- to and Ilu(tl) - all 2 ~ / 3 6 . The above-mentioned assertion and property (7.7) imply that ~(u(t)) tends to zero when t---> +o0; the same holds true for Ilu(t) - all, due to property (7.6). REFERENCES 1. AUBIN J. P., Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions. Math. Analysis and Applications, Part A, Adv. Math. Suppl. Studies (Edited by L. NACnBIN), Vol. 7A, pp. 159-229, Academic Press, New York (1981). 2. Auan~ J. P. & CELLINAA., Differential Inclusions, Springer, Berlin (1983). 3. AUBIN J. P., CELLINAA. & NOHEL J., Monotone trajectories of multi-valued dynamical systems, Annali Mat. pura appl. 115, 99-117 (1977). 4. BRI~zls H., Operateurs Maximaux Monotones et Semi Groupes de Contractions dans les Espaces de Hilbert, NorthHolland, Amsterdam (1973). 5. GILBARGD. & TRUDINGERN. S., Elliptic Partial Differential Equations of Second Order, Springer, Berlin (1977). 6. HADDAD G., Monotone trajectories of differential inclusions and functional differential inclusions with memory, Israel J. Math. 39, 83-100 (1981). 7. HESTENES, M. R., Optimization Theory, Wiley, New York (1975). 8. LABORDEP., Etude d'un probl~me d'6volution non-monotone, Ann. Fac. Sci. Toulouse 5, 1-14 (1983). 9. LABORDEP., Analyse math6matique de probl/~mes en th6orie de la plasticit6, Thesis, Universit6 de Bordeaux I (1984). 10. LAaOmaE P., Analysis of the strain-stress relation in plasticity with non-associated laws, Int. J. Engng Sci (to appear). 11. MIGNOTF., Controle darts les in6quations variationnelles elliptiques, J. funct. Analysis 22, 130--185 (1976). 12. MOREAUJ. J., Evolution problem associated with a moving convex set in a Hilbert space, J. diff Eqns 26, 347374 (1977). 13. ROCKAFELLARR. Z., Convex Analysis, Princeton University Press (1970). 14. ZARANTONELLOE., Projections on convex sets, in Contributions to Non Linear Functional Analysis. Symposium, Madison, 1971, Academic Press, New York (1971).