Slow and Quasi-slow solutions of differential inclusions

jVonlincar Analysu. Theory. Pnntcd III Great Britain.

Yerhodr

& Applicononr.

Vol.

Il.

No. 3. pp. 367-377.

0362-54Xi87 Pergamon

1987

SLOW AND QUASI-SLOW SOLUTIONS INCLUSIONS* M. Dipartimento di Matematica,

53.00 + .@I Journals Ltd.

OF DIFFERENTIAL

FALCONE

Universita di Roma, P.A. Moro 2, 00185 Roma, Italy

and P. SAINT-PIERRE CEREMADE.

Universite Paris IX-Daupinr. (Received

11 December

Kqv words and phrases: Differential

Place du Xlarechal de Lattre de Tassigny, 75775 Paris. France 1985; receiued for publication 11 April 1986)

inclusions, slow solution. viable trajectories,

tangent cone.

1. INTRODUCTION WE PRESENT

to the Cauchy

in this paper problem

some sufficient

conditions

i.(r) E Wr))

for the existence

of a slow viable solution

a.e. in t E [0, +r-)

x(O) = xg E K

(1.1) vt E [O, +=)

x(t) E K

i

where K is a closed convex subset of a finite dimensional vector space X and F : K-, X is a bounded upper semi-continuous set-valued map with convex compact images. A slow solution of (1.1) is defined to be an absolutely continuous function satisfying i.(t) = m(x(t))

a.e. in I E [0, +=)

x(O) = x0 E K

(1.2) v’t E [O, +x)

x(t) E K

I

where m(x) denotes the element of minimal norm in the set R(x) = F(x) fl T&c) and T&) the tangent cone to K at the point x, defined as follows

is

(1.3) A necessary and sufficient that (see Haddad [6]) (V) F(x) n T&)

# 0

condition

for the existence

Yx E K (viability

of a solution

to (1.1) for all x0 E K is

condition).

Since the set-valued map x+ R(x) = F(x) rl TK(x) is not sufficiently regular in general, need to provide stronger assumptions to prove the existence of a slow viable solution. l This work was partially supported by the Minis&e des Relations Exttrieures the first author was staying at CEREMADE.

367

Fraqais

we

and was completed when

368

M. F&CONE and P. SAM-PIERRE

After some preliminary results and definitions we introduce in the next section the concept of quasi-slow solution associated to (1.1). We then show that under appropriate assumptions there exists at least one quasi-slow solution of (1.1). In Section 3 we study the case when F is continuous, pointing out that it might not exist a slow solution and we give different sufficient conditions which imply the existence of a slow solution. In Section 4 we extend the concept of slow solution to viable solution whose derivative minimizes a given function y and we show that same conditions imply existence of solution to these problems.

2. A SUFFICIENT

CONDITION

We shall always assume that X is a finite dimensional vector space and K is a closed convex subset of X. Let F be a bounded set-valued map from K to X with nonempty convex compact values such that R(x) = F(x) n T&)

Vx E K.

# 0

(2.1)

Then, since R(x) is a convex compact subset of X, we can define (2.2)

m(x) = pro] a~.~) 10) the unique element of minimal norm in R(x). Let us consider the set-valued map M(x) = n co m(x + EB) 00

(2.3)

where B is the unit ball of X m(x + EB) = {m(x’) : x’ E (x + EB) f-~K} and CiA denotes the closed convex hull of A. Definition.

A

quasi-slow

solution

of

(1.1)

is

an

absolutely

continuous

function

x( .) : [0, +x) + X such that i(t) E MC@)

a.e. t E [0, +p)

x(O) = x0 E K x(t) E K

(2.4) Vt E [O, +a).

1

The set-valued map M defined in (2.3) is the smallest upper semi-continuous set-valued map with convex compact values containing m(e). (See Henry [8]). Therefore, the following existence results holds true PROPOSITION 2.1. Let F : K+ X be a bounded set-valued map with nonempty convex compact values. Then there exists at least one quasi-slow solution of (1.1).

Slow and quasi-slow solutions of differential inclusions

It is obvious that any slow solution is also a quasi-slow solution. The converse, is not true as the following example shows:

369

in general,

Example. Let X = Y = W*and K = [0, l] x [0,11. Let F be a U.S.C. set-valued map defined on K by: ifxq #O

((19 0)) ~(xl~~2)

P,

=

11x [O,11

andxl

if(xl,d

# 1

= Cl,11

ifxt=O orx,=l. [0,1]~[0,1]~CyER2:y~+y2~1} I We have, for any x E [0, l] x {0}, T,(x) = Cy : y2 3 0) then, for such an x, T,(x) > F(x) and m(x) = (l/2, l/2). On the other hand, T,((l, 0)) = b E Iw’: y, S 0 and y, 3 0) which implies 0)) r-~F((1, 0)) = [(O, 1)).

T&l,

By definition, for any x E [0, l] x {0}, we have M(x) = co[(1/2,1/2),

(1, O)] = cy E lR*: y, + y2 = 1,1 S y, S 1).

Note that at these points M(x) fl TK(x) # {m(x)} and in this case a slow solution could not exist for any starting point x0 E [0, 1) X (0). In fact,

x(t) =

(4 + t, 0)

forOSttl-xy

(1, t - 1 -xi)

for 1 - x7 < t c 2 + xp

fort>2+xy (111) 1 is the unique slow quasi-solution defined on [0, +=) since C&l,

1)) n

F((1, 1)) = (01.

It is not a slow solution since i(t)

However,

= (1,O) f m(x(t))

= (l/2,

l/2)

Vt E [O, 1 - x1].

we point out the following result.

2.2. Let F be a set-valued map from K to X as defined in proposition Assume the condition

PROPOSITION

(s) M(X) n w4

= bw)

2.1.

‘ix E K

be satisfied. Then there exists at least one slow solution of (1.1). Proof. By proposition 2.1, there exists a quasi-slow solution .?( -) of (1.1) and, since X(t) E K, Vt E [0, +m), it verifies g(t) = lim h-0

qt + h) - 2(t) h

E mw)

a.e. in t E [0, +a).

(2.5)

,M. F.ucom

370

and P. SAINT-PIERRE

Condition (S) implies: a.e. in t E [O. +=I

f(r) = m(Y(r)) so that i( .) is a slow solution.

n

Let us observe that condition (S) implies that the set of quasi-slow solutions coincides with the set of slow solutions since (2.5) holds for any quasi-slow solution. We recall that M(x) is reduced to m(x) whenever m is continuous at x. 3. SLOW

SOLUTIONS

FOR

CONTINUOUS

SET-VALUED

MAPS

When F is continuous, it is quite easy to prove the local existence of slow trajectories at the interior of K when the latter is not empty. Indeed, for any x E K, R(x) = F(x), since TK(x) = X, and m( .) is continuous (see Berge [4]). So we can apply the Peano theorem to the Cauchy problem: i(f) = m@(t)) 1

x(0) = xg E k.

On the boundary of K, the tangential condition R(x) # & becomes restrictive and the continuity, even the lower semi-continuity of R(v), may be lost. However, we have the following result. 3.1. Let K be a closed convex subset of X, and F: K+ X be a bounded continuous set-valued map with nonempty convex compact values. Assume that, at a point x E K, one of the following conditions holds:

THEOREM

(C,) 3~> 0, 36 > 0 such that for any x’ E B(x, 6) n K. yB C F(L)

- T&d);

or (C,) k z 0

and F(x) n Int TK(x) f 0.

Then (S) M(x) fl T/G)

= {m(x)>.

Proof. Condition (C,) or condition (C,) implies that the set-valued map R( .) is lower semicontinuous at x (see Aubin and Cellina [2], Aubin [l]). Then the map x--, ]/m(x)]1 is upper semi-continuous, in the sense of functions, at x: V’E> 0, there exists r] > 0 such that for any x’ E B(x, q) llW’)ll s

IlW>ll+ E.

(3.1)

This implies, by the definition of M, that

llzll=G Ilm)ll

vz E M(x)n T,(X).

(3.2)

371

Slow and quasi-slow solutions of differential inclusions

Since F is upper semi-continuous,

convex

ll~(~)ll 6 //~/I Since the projection

valued. ‘M(x) is contained in F(x) and (3.3)

vz f M(X) n z-,(X).

is unique, (3.2) and (3.3) imply that

(S)M(x) l-l TK(X)= {m(x)).

n

By theorem 3.1 and proposition 2.2 we can deduce an interesting result which gives a different sufficient condition for the existence of a slow solution when the images of F(x) are “strictly convex” in the following sense. Definition. F is “strictly convex” at a point x if F(x) is convex and VY E dF(x), y is an extremal point of F(x) (Rockafellar [9]). THEOREM 3.2. Let K and F as defined in theorem

3.1. If

(C,) F is “strictly convex” at any point x E aK, then there exists at least one slow-solution of (1.1). Before giving the proof of this theorem, we need the following preliminary

result.

LEMMA 3.3. Let F be lowerOsemi-continuous set-valued map at a point x with nonempty convex compact values. Assume F(x) # 0. Then, for any Y E F(x) and for any y> 0 such that

B(y 7 y) c

&> .

(3.4)

there exists 17> 0 such that t’x’ E B(x. n), B(Yt r) C F(X,). Proof of lemma 3.3. Since F(x) is compact. from (3.4) there exists E> 0 such that B(y. y + E) C F(x) and there exists a finite family of points ,vjE aB(y, y + E) such that aB(y, Y + E) c

Since F is lower semi-continuous

U qy,, 42). iEl

at x and yi E F(x)

Vi E I, there exists vi > 0 such that Vx’ E B(x, vi), B(Y;, &/2) n F(d) f 0. Let

Then, Vx’ E B(x, q), Vi E I, there exists zi E B(y,, &/2) n F(d). Let us consider now the convex hull of points 2;: c = co{z;}iE[.

(3.5)

372

M. F.ucom

and P. SAINT-PIERRE

We have just to prove that C contains B(y, y). Assume that it is not true: there exists z E B(y, y), z # y, such that z $ C. Then, from the strict separation theorem, it is easy to show that there would exist a point y’ E as(Y, y + E), for instance either y’ = y + (y + E)(z/]~z~])or y’ = y - (y + E)(z/]~z~]),such that B(y’, E) n c = 0.

(3.6)

But from (3.5), there would exist some index i. E I for which Y’ E B(Y,,, 42) C B(Y’, 4 and zz, E which is in contradiction

B(Yio

7 E/2)

n

F(x’)

with (3.6) since zio E C.

c

B(Y’7

El

H

We can now prove theorem 3.2. Proof of theorep 3.2. Assume that the condition (S) fails at at any point x E K). Then there would exist yr and y2, y, # y,, Since F is upper semi-continuous, convex valued, then M(x) property of F at x implies that F(x) has a nonempty interior,

a point x E X (it always holds which belong to M(x) fl T&x). C F(x) and the strict convexity moreover

lY1,Y*] C R-4 fl TK(X). Let y E ]yr, y2[ and y’ > 0 such that &Y, Y’) C F(x). Since F is lower semi-continuous r]t > 0 such that:

with convex compact values, by lemma 3.3, there exists

&Y, V) C Q’)

t/x’ E B(x, 7,).

On the other hand, since TK( .) is lower semi-continuous,

there exists ~7~> 0 such that

‘dx’ E B(x, q2).

Then, if v = min{n,, q2}, we have: ; &O, 1) C &y, y’) - TK(x’) C F(x’) - &(x’). Choosing y = y’/2 and 6 = 7, condition (C,) of theorem 3.1 is satisfied and it implies that M(x) fl TK(x) has a unique element which is opposite to our initial assumption. Condition (S) holds at any point x E K and, from proposition 2.2, there exists at least one slow solution to (1.1). H Remark 1. The main interest of this last result concern the fact that it extends the existence theorem 3.1 to the case when F(x) or F(x) n TK(x ) re d uce to a single point belonging to the

boundary of TK(x) (see Figs l-3).

373

Slow and quasi-slow solutions of differential inclusions

Fig. 2. Condition (Cz) is satisfied.

Fig. 1. Condition (C,) is satisfied.

Fig. 3. Condition (C,) is satisfied but (C,) and (Cl) are not.

Remark

2. When we know that the set-valued map R is lower semi-continuous remark due to Frankowska*.

we can use a

Let us consider the set-valued map H defined by H(x) = Ilm(x)llB II F(x) where B is the unit ball of X. H is an upper semi-continuous set-valued map with convex compact values since the map x-+ Ilm(x)ll is upper semi-continuous in the sense of functions. Then, the viability theorem implies the existence of a viable solution x( *) of the differential inclusion: 49 E H($)) i

x(0) = x0 E K

Since for almost t E [0, +a), Ik(t)ll s Ijm(x(t))ll. Therefore

i(t) E T&(t))

i(t) = m(x(t)) l

CEREMADE,

a.e. in t E [0, +=)

Paris IX-Dauphine.

II H(x(t))

a.e. in t E [0, +=).

then

i(t) E R(x(r))

and

M. FALCONE and P. SAINT-PIERRE

371

Conditions

(C,) and (Cl) imply that the set-valued map R is lower semi-continuous. 4.

y-MINIMAL

VIABLE

SOLUTIONS

The concept of slow solution can be extended to the case of viable solution whose derivative map minimizes a given function. Let y be a convex, continuous, inf-compact function from X to R such that y(z) < +=.

(4.1)

Under assumption of theorem 3.1. if either condition (C,) or condition

(Cl)

.;gR_:$ THEOREM

4.1.

holds, there exists a viable solution of the differential

inclusion: a.e. f E [O. +x)

40 E F(x(r)) x(0) =

x0

E K

(4.2) 1 J

such that y(i(t)) = Proof.

min

: E R(.r(r))

a.e. f E [O. f=).

y(z)

Let G be the set-valued map defined bY

(

G(x) = Y E Xsuch that y(y) s ,FRf, Let us show that G is upper semi-continuous inf

: E R(x)

y(z)].

with convex compact

values. Let q(x) =

y(z). We consider a sequence (x,,,Y,J E Graph G uhich converges to (,c,_?). For any x E X, for any (YE E such that q(x) < CY, there exists z, E R(x), y(z,) < C-Y.

Since y is upper semi-continuous, there exists a neighborhood A(z,) such that Vz’ E A(z,), my < a. Under condition (C,) or (Cz) of theorem 3.1. the set-valued map R is lower semicontinuous. There exists a neighborhood N(x) of x such that Vx’ E N(x),

3z” E A(z,)

such that z” E R(x’)

then Vx’ E N(x),

&x’) s Ty(Z”)< (Y

and q is upper semi-continuous. Since I4Yn) s &?I)

Vn E K

and y is lower semi-continuous y(y) C lim inf y(yn) 5 limyyp dx,) ?I-= then (Z,?) E Graph G which is closed.

s q(X)

Slow and quasi-slow solutions of differential inclusions

375

Since y is inf-compact and from (4. l), Vx E X, G( x ) IS included in a fixed compact set. Then G is upper semi-continuous. Let H(x) = G(x) rl F(x). The set-valued map H is also upper semi-continuous and Vx E K, G(X) n R(x) # 0. Then VX E K, H(X) n T,( x ) 1snonempty. The viability condition (V) implies the existence of a viable solution to the differential inclusion

X(t) E f+(r))

a.e. t E [0, +=)I L

x(0) =x0 E K

(4.3)

I

and any viable solution of (4.3) satisfies y(?;(t)) =

min y(z). zENx(r))

H

The “strictly conoex” case

The lower semi-continuity property of the set-valued map R is a crucial condition to get the existence of slow solution or y-minima1 viable solution. We shall prove that in the strictly convex case, this property holds true.

LEMMA 4.2. Let F be a continuous set-valued map with strictly convex compact values from K to X, where K is such that the set-valued map x + TK(x) is lower semi-continuous with closed convex values. Then the set-valued map x+ R(x) = F(x) n TK(x) is lower semi-continuous. Prpof. (a) If R(x) is not a singleton, then, since F(x) is strictly convex, there exists y E F(x) fl TK(x) and, applying the demonstration of theorem 3.2, the condition (C,) of theorem 3.1 holds which implies the lower semi-continuity property of R. (See Aubin and Cellina [2].) (b) Let us assume that R(x) = {z}. If R is not lower semi-continuous at the point x, there would exist a neighborhood A(z) such that VN(x), neighborhood of x, 3x’ E N(x) n K such that R(x’) fl A(z) = 0.

(4.4)

The lower semi-continuity of TK implies that, since z E TK(x) there exists a sequence E,,which converges to zero such that Vx’ E

qx,

E,)

n K,

Then, from (4.4), we can built a sequence x, E B(x, E,,) rl K such that: R(x,)

n A(z) = 0

Vn E N.

(4.5)

Let yn E R(x,) and z, E TK(x,) n B(z, l/n). Since F is upper semi-continuous with convex compact values, there exists a subsequence, still called y,, which converges to an element YE F(x), Y $A@).

The strict convexity of F(x) implies that ]y, z[ C R(x). Let us choose a point z’ E ]y, z[ n A(z). There exists a ball B(z’, Ed) contained in F(x) n A(z).

376

M. FALCONEandP.SAINT-PIERRE

Then, let us consider the family of segments [yn, z,]. Vn E N,

[Yrl) znl =

T&n).

Since (y,, z,) converges to (y, z), VE < co, there exists N large enough such that Vn 2 N, Since F is lower semi-continuous

[Yn,

znl fl Nz’, &If

0.

and z’ E 6((x), there exists E < co and Nz such that B(z’, E) C F(x,).

V’n2Nz, Then,

R(x,) n B(z’, E) f 0 j which is in contradiction

R(x,) f~ A(z) f 0

with (4.5). So (4.4) never holds and R is lower semi-continuous.

n

We can now state the following result: 4.3.Let K be a compact subset of Y and F: K ---, X be a continuous set-valued map with nonempty strictly convex compact values. Let y be a map from X to R defined as before. There exists a viable solution of (4.2) such that: THEOREM

Yw) = 2E?%)) fiz)

a.e. t E [0, +=)

By lemma 4.2, the set-valued map R is lower semi-continuous which allows to use the proof of theorem 4.1. We can generalize this type of selection of viable trajectories to the research of solutions which satisfy the condition: (4.6)

where q is a continuous map from X x X to R such that (i) Vz E X,

Y+ T(Y? z)

is convex

(ii) Vy E X,

*--, (P(Y,z)

is concave

(4.7)

(iii) ;E$ q(z, z) < 0. 4.4. Under assumptions of theorem 4.1 or theorem 4.3 there exists a viable solution of (4.2) such that (4.6) holds.

THEOREM

We apply the proof of theorem

(4.1) or theorem (4.3) to the set-valued map G defined by:

G(x) = [ y E Xsuch that Z;u~I, &y, z) < 0 . I

The minimax theorem implies the existence of an element jj E R(x) such that

Slow and quasi-slow solutions of differential inclusions

377

REFERENCES 1. AUBIN J. P., Slow and heavy trajectories of controlled problems. Smooth viability domains, in ~Multifincrion.s and Inregrunds (Edited by G. SUINETIT), Lecmre Notes in Machemarics 1091.Springer, Berlin (1984). 2. AUBIN J. P. & CELLINA A., Diflerenfiul Inclusions, Springer, Berlin (1984). 3. AUFHNJ. P. & FRANKOWSKAH., Heavy viable trajectories of controlled systems, Ann. Inst. H. PoincarP, Anulyse Non Linkuire 2, 371-395 (1985). 4. BERGE C., Espuces Topologiques, Fonctions Mulrivoques, Dunod, Paris (1966). 5. CORNETB., Existence of slow solutions for a class of differential inclusions. /. murk Analysis Applic. 96. 130-147 (1983). 6. HADDAD G., Monotone viable trajectories for functional differential inclusions, 1. difi Eqns 42, l-24 (1981). 7. HENRY C., Differential equations with discontinuous right hand-side for planning procedures, J. econ. Theor. 4, 545-551 (1972). 8. HENRY C., An existence theorem for a class of differential equations with multivaiued right-hand side, J. math. Analysis Applic. 41, 17S186 (1973). 9. ROCKAFELLARR. T., Convex Analysis, Princeton University Press, Princeton (1970).