OF NONCONVEX DIFFERENTIAL INCLUSIONS

A VIABILITY RESULT FOR A CLASS

OF NONCONVEX DIFFERENTIAL INCLUSIONS

AURELIAN CERNEA

We prove the existence of viable solutions to the Cauchy problem x ∈ F(x) + G(t, x),x(0) =x0,x(t)∈K, whereK⊂Rⁿ is a locally compact set,G(·,·) is a continuous set-valued map andF(·) is an upper semicontinuous set-valued map contained in the Fr´echet subdifferential of aφ-convex function of order two.

AMS 2000 Subject Classification: 34A60.

Key words: viable solution,φ-convex function of order two, nonconvex differential inclusion.

1. INTRODUCTION

Consider a multifunction F : Rⁿ → P(Rⁿ) that defines the Cauchy problem

(1.1) x∈F(x), x(0) =x₀.

In the theory of differential inclusions the viability problem consists in prov- ing the existence of viable solutions, i.e., ∀t, x(t) ∈ K, to the Cauchy pro- blem (1.1).

The viability problem was first solved by Nagumo [11] for differential equations (i.e., F is a single valued map). Nagumo’s theorem was forgotten and rediscovered several times. Under the assumptions that F is an upper semicontinuous nonempty convex compact valued multifunction andK is lo- cally compact, Haddad [10] proved that a necessary and sufficient condition for the existence of viable trajectories starting fromx₀ ∈K of problem (1.1) is the tangential condition∀x∈K,F(x)∩T_xK =∅, whereT_xKis the contingent cone toK at x∈K.

Rossi [12], proved the existence of viable solutions to problem (1.1) re- placing the convexity conditions on the images onF by

(1.2) F(x)⊂∂V(x) ∀x∈K,

REV. ROUMAINE MATH. PURES APPL.,52(2007),1, 1–8

where∂V is the subdifferential in the sense of Convex Analysis of a proper con- vex functionV. Condition (1.2) was first introduced by Bressan, Cellina and Colombo [4] for a differential inclusion problem without viability constraints.

The result in [12] was improved in [7] by assuming that

(1.3) F(x)⊂∂_FV(x) ∀x∈K,

where∂_FV is the Fr´echet subdifferential of aφ-convex function of order twoV. In a recent paper, Duc Ha [9] showed the existence of viable solutions to the problem

(1.4) x ∈F(x) +G(t, x), x(0) =x₀ ∈K, x(t)∈K, ∀t∈[0, T], whereG(·,·) is a continuous set-valued map,K ⊂Rⁿ is locally compact and F(·) is as in [12].

In [3] the convexity assumption on the functionV in [9] is relaxed, in the sense thatV is assumed to be locally Lipschitz uniform regular and F(x) ⊂

∂_CV(x) ∀x∈K, where∂_CV is Clarke’s generalized gradient of the map V. The aim of the present paper is to prove the existence of viable solutions to the problem (1.4) withF(·) as in [7]. On one hand, our results extends the result in [7] to the more general problem and, on the other hand, our result provides an alternative improvement of the result in [9]. Unlike the approach in [3] we do not assume thatV(·) is locally Lipschitz and we use the Fr´echet subdifferential ofV(·). Note that one always has ∂_FV(x)⊂∂_CV(x).

The proof of our main result follows the general ideas in [7] and [9].

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

2. PRELIMINARIES

We denote by P(Rⁿ) the set of all subsets of Rⁿ and by R₊ the set of all positive real numbers. For > 0 we putB(x, ) ={y ∈Rⁿ;y−x< }. WithB we denote the unit ball in Rⁿ. By cl(A) we denote the closure of the setA ⊂Rⁿ, by co (A) the convex hull of A and putA= sup{a;a∈A}. We recall that onP(Rⁿ) the generalized Hausdorff metric is defined by

d_H(A, B) = max{d^∗(A, B),d^∗(B, A)}, d^∗(A, B) = sup{d(a, B);a∈A}, where d(a, B) = inf_b∈Ba−b and · is the Euclidian norm onRⁿ.

Let Ω⊂ Rⁿ be an open set and letV : Ω → R∪ {+∞}be a function with domainD(V) ={x∈Rⁿ;V(x)<+∞}.

Definition2.1. The multifunction∂_FV : Ω→ P(Rⁿ) defined by∂_FV(x) =

α ∈ Rⁿ,lim inf

y→x

V(y)−V(x)−α,y−x y−x ≥ 0

if V(x) < +∞ and ∂_FV(x) = ∅ if

V(x) = +∞is called theFr´echet subdifferentialofV. PutD(∂_FV) ={x∈Rⁿ;

∂_FV(x)=∅}.

According to [8] the values of ∂_FV(·) are closed and convex.

Definition 2.2. Let V : Ω → R∪ {+∞} be a lower semicontinuous function. We say that V is φ-convex of order 2 if there exists a continuous mapφ_V : (D(V))²×R² →R₊ such that for everyx, y ∈D(∂_FV) and every α∈∂_FV(x) we have

(2.1) V(y)≥V(x) +α, x−y −φ_V(x, y, V(x), V(y))(1 +α²)x−y². In [6], [8] there are several examples and properties of such maps. For example, according to [6], ifK ⊂R² is a closed and bounded domain whose boundary is aC² regular Jordan curve, then the indicator function ofK,

V(x) =I_K(x) =

0, ifx∈K +∞, otherwise, isφ-convex of order 2.

In what follows we assume

Hypothesis2.3. i)K ⊂Rⁿ is locally compact.

ii) G(·,·) : [0, a]×Rⁿ → P(Rⁿ), a > 0,is a continuous set-valued map with compact values.

iii)F :Rⁿ→ P(Rⁿ) is upper semicontinuous with compact values.

iv)There exists a proper lower semicontinuousφ-convex function of order twoV :Rⁿ→R∪ {+∞} such that

F(x)⊂∂_FV(x) ∀x∈K.

v)For any (t, x)∈[0, a]×K, the tangential condition lim inf

h→0

1

hd(x+h[F(x) +G(t, x)], K) = 0, holds.

Finally, we recall the following result proved in [9].

Lemma2.4 ([9]). Assume that

i) K ⊂ Rⁿ is nonempty, x₀ ∈ K, and there exists r > 0 such that K₀ :=K∩B(x₀, r) is compact;

ii) P(·,·) : [0, a]×Rⁿ → P(Rⁿ) is an upper semicontinuous set-valued map with nonempty compact values;

iii)for any (t, x) ∈[0, a]×K, the tangential condition lim inf

h→0

1

hd(x+hP(t, x), K) = 0 holds.

Let T∈(0,min{a,_M^r₊₁}), where M := sup{P(t, x); (t, x)∈[0, a]×K₀}. Then for any ε ∈ (0,1), any set N ={t_i; 0 = t₀ <· · · < t_ν = a} and any u₀ ∈ P(0, x₀), there exist a set N = {t_i; 0 = t₀ < · · · < t_ν = T}, step functionsf(·),z(·) andx(·) defined on[0, T]such that for every i∈ {1, . . . , ν} the conditions below hold.

1){t₀, . . . , t_k(i)} ⊂ {t₀, . . . , t_i}, where k(i) is the unique integer such that k(i)∈ {0,1, . . . , ν−1} andt_k(i)≤t_i ≤t_k(i)+1.

2) 0< t_j+1−t_j ≤α for allj∈ {0,1, . . . , i−1}, where α:=εmin{1, t₁− t₀, . . . , t_ν−t_ν−1}.

3) p(0) = u₀, p(t) = p(θ(t))∈P(θ(t), x(θ(t))) on [0, t_i], where θ(t) = t_j ift∈[t_j, t_j+1), for allj∈ {0,1, . . . , i−1} and θ(t_i) =t_i.

4) z(0) = 0, z(t) = z(t_l+1) if t ∈ (t_l, t_l+1], l ≤ i−1 and z(t) ≤ 2ε(M + 1)T.

5) x(t) = x₀+_t

0 p(s)ds+z(t) ∀t ∈ [0, t_i], x(t_j) = x_j ∈ K₀ and x_j − x_j ≤(M+ 1)|t_j −t_j|for j, j ∈ {0,1, . . . , i}.

3. MAIN RESULT Our main result is as follows.

Theorem 3.1. Assume that Hypothesis 2.3 holds. Then for all x₀ ∈K there existsT ∈(0, a) such that problem(1.1) admits a solution on [0, T].

Proof. Let r > 0 be such that K₀ := K∩B(x₀, r) is a compact set in Rⁿ. Let M, T >0 be such that F(x)+G(t, x) ≤M ∀(t, x)∈[0, a]×K₀ andT ∈(0,min{a,_M^r₊₁}). LetN₀ :={0, T} and ε_m = ₂¹_m,m= 1,2, . . . .

First, the uniform continuity ofG(·) on the compact set K₀ ensures the existence ofδ_m>0 such that∀t, t∈[0, T] and∀x, x ∈K₀if(t, x)−(t, x) ≤ (M+ 2)δ_m, then d_H(G(t, x), G(t, x))≤ε_m.

Apply Lemma 2.4 withP(t, x)≡F(x) +G(t, x) and ε_m,m= 1,2, . . . . Therefore, for every m = 1,2, . . . there exist a set N_m ={t^m_i ; 0 =t^m₀ <

· · · < t^m_νm = T}, and step functions f_m(·), g_m(·), z_m(·) and x_m(·) defined on [0, T] with the following properties:

i)N_m ⊂N_m+1 ∀m≥0;

ii) 0 < t^m_i+1 −t^m_i ≤ α_m for all i ∈ {0,1, . . . , ν_m −1}, where α_m :=

ε_mmin{1, δ_m, t^m−1₁ −t^m−1₀ , . . . , t^m−1_νm−1 −t^m−1_νm−1−1};

iii) g_m(t) =g_m(θ_m(t))∈ G(θ_m(t), x_m(θ_m(t))) and f_m(t) = f_m(θ_m(t))∈ F(x_m(θ_m(t))) on [0, T], whereθ_m(t) =t^m_i ift∈[t^m_i , t^m_i+1], for all i∈ {0,1, . . . , ν_m−1}and θ_m(T) =T;

iv) z_m(0) = 0, z_m(t) = z(t_i+1) if t ∈ (t_i, t_i+1], 0 ≤ i ≤ ν_m −1 and z_m(t) ≤2ε_m(M+ 1)T;

v)x_m(t) =x₀+_t

0[f_m(s) +g_m(s)]ds+z_m(t),x_m(θ_m(t))∈K₀ ∀t∈[0, T], and

x_m(t^m_i )−x_m(t^m_j ) ≤(M+ 1)|t^m_i −t^m_j | fori, j∈ {0,1, . . . , ν_m}.

Note that from the last inequality for i, j∈ {0,1, . . . , ν_m} we get (t^m_i , x_m(t^m_i ))−(t^m_j , x_m(t^m_j )) ≤(M+ 2)|t^m_i −t^m_j |.

Remark that the sequence g_m(·) can be constructed using the relative compactness property in the space of bounded functions. The proof of this fact can be found in [9]. Therefore, without loss of generality, one may assume that there exists a bounded function g(·) such that

(3.1) lim

m→∞ sup

t∈[0,T]g_m(t)−g(t)= 0.

Define q_m(t) = x₀ + _t

0[f_m(s) +g_m(s)]ds. By property iv), one has z_m (t)= 0 a.e. on [0, T] and thus q_m (t)=x_m(t) ≤M a.e. on [0, T] and the sequence q_m(·) is equicontinuous while the sequence of derivatives q_m (·) is equibounded. Therefore, a subsequence (still denoted) q_m(·) converges in the sup-norm topology to an absolutely continuous functionx(·) : [0, T]→Rⁿ while the sequence of derivatives q_m(·) converges weakly in L²([0, T],Rⁿ) to x(·). At the same time,q_m(t)−x_m(t)=z_m(t),z_m(t)= 0 a.e. on [0, T].

Hence by iv) it follows that

(3.2) x_m(·) converges uniformly to x(·),

x_m(·) converges weakly in L²([0, T],Rⁿ) tox(·).

On the other hand,g_m(·) converges pointwise a.e. on [0, T] tog(·). Then the continuity ofG(·,·) and the fact that the values ofG(·,·) are closed yield g(t)∈G(t, x(t)) a.e. on [0, T]. Further, by the properties of the sequencex_m(·) and the closedness ofK₀, we getx(t)∈K₀ ⊂K.

Definef(t) :=x(t)−g(t). By construction,f_m(t) :=x_m(t)−g_m(t) and (3.3) f_m(t)∈F(x_m(θ_m(t)))⊂∂_FV(x_m(θ_m(t))) a.e.([0, T]).

Since f_m(t) ≤ M ∀t ∈[0, T], m ≥ 1, it follows (e.g. Theorem III.27 in [5]) that there exists a subsequence (again denoted) f_m(·) such that f_m(·) converges weakly inL²([0, T],Rⁿ) to f(·).

Therefore from (3.3) and Theorem 1.4.1 (“Aubin-Cellina convergence theorem”) in [2] we obtain

f(t) =x(t)−g(t)∈coF(x(t))⊂∂_FV(x(t)) a.e.([0, T]).

Next, apply Theorem 2.2 in [6] to deduce that

(V(x(t))) =f(t), x(t)=x(t)²− g(t), x(t) a.e.([0, T]).

Thus

(3.4) V(x(T))−V(x₀) = _T

0 x(t)²dt− _T

0 g(t), x(t)dt.

On the other hand, by construction we have x_m(t) = f_i^m +g_i^m with f_i^m ∈F(x^m_i )⊂∂_F(x^m_i ). On account of Definition 2.2, one has

V(x^m_i+1)−V(x^m_i )≥ f_i^m, x^m_i+1−x^m_i −

−φ_V(x^m_i+1, x^m_i , V(x^m_i+1), V(x^m_i ))(1 +f_i^m2)x^m_i+1−x^m_i ² =

=

x_m(t)−g_m(t), _t^m_i+1

t^m_i x_m(t)dt

−

−φ_V(x^m_i+1, x^m_i , V(x^m_i+1), V(x^m_i ))(1 +f_i^m2)x^m_i+1−x^m_i ² ≥

≥ _tm

i+1

t^m_i x_m(t)²dt− _tm

i+1

t^m_i x_m(t), g_m(t)dt−

−φ_V(x^m_i+1, x^m_i , V(x^m_i+1), V(x^m_i ))(1 +f_i^m2)x^m_i+1−x^m_i ². Summing over ithe last inequalities we get

V(x_m(T))−V(x₀)≥ _T

0 x_m(t)²dt− _T

0 x_m(t), g_m(t)dt−a(m), where

a(m) =

νm

i=0

φ_V(x^m_i+1, x^m_i , V(x^m_i+1), V(x^m_i ))(1 +f_i^m2)x^m_i+1−x^m_i ². Denote

S:= sup{φ_V(x₁, x₂, y₁, y₂);x_i∈K₀, y_i ∈[V(x₀)−1, V(x₀)+1], i= 1,2}<+∞.

According to Remark 1.14 and Theorem 1.18 in [9] (or Theorem 2.1 in [6]),V(·) is continuous onD(V). So, we have the estimate

|a(m)| ≤

νm

i=0

S(1 +F(K₀)²)x^m_i+1−x^m_i ² =

=S(1 +F(K₀)²)

νm

i=0

_t^m_i+1

t^m_i x_m(t)dt ² ≤

≤S(1 +F(K₀)²)

νm

i=0

1 2^m

_tm i+1

t^m_i x_m(t)²dt≤

≤S(1 +F(K₀)²) 1 2^m

_T

0 x_m(t)²dt≤ 1

2^mS(1 +M²)T(M+ 1)², so that lim

m→∞a(m) = 0.

According to (3.1) and (3.2), we obtain

m→∞lim _T

0 x_m(t), g_m(t)dt= _T

0 x(t), g(t)dt.

Lettingm→ ∞ in (3.5), the continuity of V(·) yields lim sup

m→∞

_T

0 x_m(t)²dt− _T

0 x(t), g(t)dt≤V(x(T))−V(x(0)).

Using (3.4) we obtain lim sup

m→∞

_T

0 x_m(t)²dt≤ _T

0 x(t)²dt

and by the lower semicontinuity of the norm inL²([0, T],Rⁿ) (e.g. Proposi- tion III.30 in [5]) we find that

m→∞lim _T

0 x_m(t)²dt= _T

0 x(t)²dt, so thatx_m(·) converges strongly in L²([0, T],Rⁿ).

Hence, there exists a subsequence (still denoted) x_m(·) which converges pointwise almost everywhere to x(·). It remains to note that (x_m(t), x_m(t)− g_m(t))∈graphF(·) a.e. ([0, T]) and the closedness of graphF(·) ensures that (x(t), x(t)−g(t))∈graphF(·) a.e. ([0, T]) and the proof is complete.

Remark3.2. IfG≡0 in Theorem 3.1, then we obtain the result in [7].

If K = Rⁿ, G(·,·) is single valued and V(·) is convex, in Theorem 3.1 then we obtain the result in [1].

IfV(·) is convex in Theorem 3.1, then we obtain the result in [9].

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Received 1 March 2006 University of Bucharest

Faculty of Mathematics and Computer Science Str. Academiei 14

010014 Bucharest, Romania