A NOTE ON VIABLE SOLUTIONS FOR A NONAUTONOMOUS DIFFERENTIAL INCLUSION WITHOUT CONVEXITY

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NONAUTONOMOUS DIFFERENTIAL INCLUSION WITHOUT CONVEXITY

AURELIAN CERNEA

We prove the existence of viable solutions to the Cauchy problemx⁰ ∈F(t, x), x(0) =x0, whereF is a multifunction defined on a locally compact set M and contained in the Clarke subdifferential of a uniformly regular function.

AMS 2000 Subject Classification: 34A60.

Key words: viable solution, uniform regular function, differential inclusion, Clarke’s subdifferential.

1. INTRODUCTION

Consider a multifunction F : M ⊂ R^m → P(R^m) that defines the Cauchy problem

(1.1) x⁰∈F(x), x(0) =x₀.

In the theory of differential inclusions the viability problem consists in proving the existence of viable solutions, i.e.∀t,x(t)∈M, to the Cauchy problem (1.1).

Under the assumptions that F is an upper semicontinuous nonempty convex compact valued multifunction and M is locally compact, Haddad [5]

proved that a necessary and sufficient condition for the existence of viable trajectories starting from x₀ ∈M of problem (1.1) is the tangential condition

(1.2) ∀x∈M F(x)∩TxM 6=∅,

where TxM is the contingent cone to M atx∈M.

Rossi [7] proved the existence of viable solutions to problem (1.1) replac- ing the convexity conditions on the images on F by

(1.3) F(x)⊂∂V(x) ∀x∈M,

where ∂V is the subdifferential, in the sense of Convex Analysis, of a proper convex function V. According to Theorem 4.2 in [2] the result in [7] can be improved by assuming that

(1.4) F(x)⊂∂_CV(x) ∀x∈M,

MATH. REPORTS10(60),1 (2008), 11–16

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where ∂CV is the Clarke subdifferential (generalized gradient) of a uniform regular function V(·).

The aim of the present note is to extend this result to the case of nonautonomous problem

(1.5) x⁰ ∈F(t, x), x(0) =x₀.

We note that in [6] a similar type of result is proved for a functionV(·) whose upper Dini directional derivative coincides with the Clarke directional derivative.

The idea of the proof of our result is to use the regularizing technique in [6] and to apply the known result for autonomous problems in [2].

The paper is organized as follows. In Section 2 we recall some preliminary facts that we need in the sequel while in Section 3 we prove our main result.

2. PRELIMINARIES

Let Ω⊂R^m be a given set. A multifunction F : Ω→ P(R^m) is called (Hausdorff) upper semicontinuous at x₀ ∈ Ω, if ∀ > 0 there exists δ > 0 such that kx−x₀k< δ implies F(x)⊂F(x₀) +B, where B is the unit ball in R^m and P(R^m) denotes the set of all subsets of R^m. For > 0 we put B(x, ) ={y∈R^m;ky−xk< }.

Consider a locally Lipschitz continuous functionV :R^m→R. For every direction v ∈ R^m its Clarke directional derivative at x in the direction v is defined by

DCV(x;v) = lim sup

y→x, t→0+

V(y+tv)−V(y)

t .

TheClarke subdifferential (generalized gradient) of V atx is defined by

∂_CV(x) ={q∈R^m, hq, vi ≤D_CV(x;v) ∀v∈R^m}.

We recall that theproximal subdifferential ofV(·) is defined by

∂_PV(x) =

q∈R^m, ∃δ, σ >0 such that V(y)−V(x) +σky−xk²≥ hq, y−xi ∀y∈B(x, δ) .

Definition 2.1. LetV :R^m→R∪ {∞} be a lower semicontinuous function and let O ⊂ dom(V) be a nonempty open subset. We say that V is uniformly regular overO if there exists a positive numberβ >0 such that for all x∈O and for all ξ∈∂PV(x) one has

hξ, y−xi ≤V(y)−V(x) +βky−xk² ∀y∈O.

We say thatV is uniformly regular over a closed set K if there exists an open set O containing K such that V is uniformly regular over O.

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In [2] there are several examples and properties of such maps. For exam- ple, any lower semicontinuous proper convex function V is uniformly regular over any nonempty compact subset of its domain with β = 0; any lower-C² functionV is uniformly regular over any nonempty convex compact subset of its domain.

The next result will be used in the proof of our main result.

Lemma 2.2 ([3]). Let V :R^m → R be a locally Lipschitz function and let ∅ 6=K ⊂dom(V) a closed set. If V is uniformly regular over K then

i)∂_CV(x) =∂_PV(x) ∀x∈K;

ii) if z(·) : [0, τ]→ R^m is absolutely continuous and z⁰(t) ∈ ∂_CV(z(t)) a.e. ([0, τ]), then

(V ◦z)⁰(t) =kz⁰(t)k², a.e.([0, τ]).

We recall that the contingent cone to a set K ⊂ R^m at x ∈ K is defined by

TxK = n

v∈R^m; lim inf

h→0+

1

hd(x+hv, K) = 0 o

.

The following viability result for differential inclusions is due to Bounk- hel [2].

Theorem 2.3 ([2]). Let K ⊂ R^m be a nonempty closed set, G : K → P(R^m) an upper semicontinuous multifunction with nonempty closed values and V(·) :R^m →Ra locally Lipschitz function that is uniformly regular over K such that

G(x)⊂∂CV(x) ∀x∈K, G(x)∩TxK6=∅ ∀x∈K.

Then for any x0 ∈K there exists τ >0 such that the problem x⁰ ∈G(x), x(t0) =x0∈K, x(t)∈K ∀t≥t0

admits a solution on [t₀, τ].

In what follows we assume

Hipothesis 2.4. i) F : [0,∞)×M ⊂ R^m → P(R^m) is a bounded set valued map, measurable with respect to t, upper semicontinuous with respect tox, with nonempty closed values.

ii) There exists a locally Lipschitz function V :R^m → R∪ {+∞} uniformly regular over the locally compact set M ⊂R^m with a constant β > 0 such that

(2.1) F(t, x)⊂∂_CV(x) ∀x∈M, a.e. t∈[0,∞), (2.2) F(t, x)∩T_xM 6=∅ ∀x∈M, a.e. t∈[0,∞).

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3. THE MAIN RESULTS

Our main result is the following

Theorem3.1. Assume that Hypothesis 2.4 is satisfied. Then for every x₀ ∈M there exists a >0 such that problem (1.5)admits a solution on [0, a]

satisfying x(t)∈M, ∀t∈[0, a].

Proof.Letx₀ ∈M. SinceM ⊂R^m is locally compact, there existsR >0 such that M0:=M∩B(x0, R) is compact. Consider

L:= sup

(t,x)∈[0,∞)×R^m

kF(t, x)k,

define T := _L+1^R and taken∈Nsuch that _n¹ < T.

By regularizing the set valued map F of the Cauchy problem (1.5) we reduce the nonautonomous problem to the autonomous case (see [6]). We can find a countable collection of disjoint subintervals (a_j, b_j)⊂[0, T],j= 1,2, . . . such that their total length is less then _n¹ and a set valued map Fn defined on D := ([0, T]\S∞

j=1(a_j, b_j))×M that is jointly upper semicontinuous and F_n(t, x)⊂F(t, x) for each (t, x)∈D. Moreover, ifu(·) andv(·) are measurable functions on [0, T] such that u(t) ∈ F(t, v(t)) a.e. t ∈ [0, T] then for a.e.

t ∈ ([0, T]\S∞

j=1(a_j, b_j)) we have u(t) ∈ F_n(t, v(t)) (we refer to [8] for this Scorza Dragoni type theorem). It is obvious that all trajectories of F are also trajectories of Fn. We extendFn to the whole [0, T]×M and define

F˜n(t, x) =











Fn(t, x) ift∈[0, T]\S∞

j=1(aj, bj) F_n(a_j, x) ifa_j < t < ^a^j^+b₂ ^j

F_n(b_j, x) if ^a^j^+b₂ ^j < t < b_j Fn(aj, x)∪Fn(bj, x) ift= ^a^j^+b₂ ^j.

It is easy to see that ˜Fn(·,·) still satisfies the tangential condition (2.2). On the other hand, according to Lemma 4 in [6], ˜Fn(·,·) is upper semicontinuous on [0, T]×M.

By extending the state space from R^m to R×R^m we can reduce our problem to the autonomous case. For every (t, x)∈[0, T]×M we define

V˜(t, x) =t+V(x).

Obviously, ˜V(·,·) is a locally Lipschitz uniformly regular function over [0, T]×

M and (1, v)∈∂CV˜(t, x) if and only ifv∈∂CV(x) for all (t, x)∈[0, T]×M. At the same time, standard arguments show that (1, v)∈T_(t,x)([0, T]×M) if and only if v∈T_xM. Therefore, conditions (2.1) and (2.2) imply that

(3.1) (1,F˜n(t, x))⊂∂CV˜(t, x) ∀(t, x)∈[0, T]×M,

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(3.2) (1,F˜n(t, x))∩T_(t,x)([0, T]×M)6=∅ ∀(t, x)∈[0, T]×M.

Thus, applying Theorem 2.3, we obtain the existence of an absolutely continuous function xn(·) : [0, a]→R^m,a∈(0, T), that satisfies

(1, x⁰_n(t))∈(1,F˜n(t, xn(t)))∩∂CV˜(t, xn(t)) a.e.[0, a], xn(0) =x0

and

(t, x_n(t))∈[0, T]×M ∀t∈[0, a].

It follows that xn(·) verifies

(3.3) x⁰_n(t)∈F_n(t, x_n(t))∩∂_CV(x_n(t)) a.e.[0, a], x_n(0) =x₀ and

(3.4) xn(t)∈M ∀t∈[0, a].

Therefore, from (3.3) we have

(3.5) kx⁰_n(t)k ≤L.

On the other hand, by (3.4), graph(xn(·)) is contained in [0, T]×M andxn(·) also is a solution to the inclusion (1.5) except for a set (say)E_nof measure not exceeding ¹_n for eachn∈N. Hence, by (3.5) and Theorem III.27 in [4] there exists a subsequence (again denoted by x_n(·)) and an absolutely continuous function x(·) : [0, a]→R^m such that

x_n(·) converges uniformly tox(·),

x⁰_n(·) converges weakly in L²([0, T],R^m) to x⁰(·).

Since graph(∂CV) is closed, by (3.3) one has

(3.6) x⁰(t)∈∂_CV(x(t)) a.e.[0, a].

We apply Lemma 2.2 and deduce that

(V(x(t)))⁰=hx⁰(t), x⁰(t)i=kx⁰(t)k² a.e.[0, a].

Therefore, V(x(a))−V(x₀) = Ra

0 kx⁰(t)k²dt. On the other hand, from (3.3) we deduce that

Z a 0

kx⁰_n(t)k²dt= Z a

0

(V ◦x_n)⁰(t)dt=V(x_n(a))−V(x₀).

Hence, by the lower semicontinuity of V, we get

n→∞lim Z a

0

kx⁰_n(t)k²dt=V(x(a))−V(x0) = Z a

0

kx⁰(t)k²dt,

so {x⁰_n(·)} converges strongly in L²([0, a],R^m). Hence, there exists a subsequence (still denoted) x⁰_n(·) which converges pointwise almost everywhere to x⁰(·). From (3.3) and the fact that graph(F) is closed we have

x⁰(t)∈F(t, x(t)) a.e.[0, a]

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and from (3.4) we obtain that ∀t∈[0, a], x(t)∈M.

Remark3.2. The result in Theorem 3.1 is still valid if instead ofR^m we work on a real Hilbert space X. The main tool in the proof this time is an infinite dimensional version of Theorem 2.3, namely, Theorem 2.3 in [3].

REFERENCES

[1] J.P. Aubin and A. Cellina,Differential Inclusions. Springer, Berlin, 1984.

[2] M. Bounkhel,Existence results of nonconvex differential inclusions. Port. Math. (N.S.) 59(2002), 283–309.

[3] M. Bounkhel and T. Haddad, Existence of viable solutions for nonconvex differential inclusions. Electron. J. Diff. Equations2005, No. 50, 10 pp.

[4] H. Brezis,Analyse fonctionelle, th´eorie et applications. Masson, Paris, 1983.

[5] G. Haddad, Monotone trajectories of differential inclusions and functional inclusions with memory. Israel J. Math.39(1981), 83–100.

[6] Z. Kannai and P. Tallos,Viable solutions to nonautonomous inclusions without convexity. CEJOR Cent. Eur. J. Oper. Res.11(2003), 47–55.

[7] P. Rossi, Viability for upper semicontinuous inclusions without convexity. Differential Integral Equations5(1992), 455–459.

[8] T. Rzezuchowski, Scorza-Dragoni type theorems for upper semicontinuous multivalued functions. Bull. Acad. Polon. Sci. S´er. Sci. Math.28(1980), 61–66.

Received 8 January 2007 University of Bucharest

Faculty of Mathematics and Computer Sciences Str. Academiei 14

010014 Bucharest, Romania