Existence and Controllability for Stochastic Evolution Inclusions of Clarke's Subdifferential Type

In this paper, we investigate a class of stochastic evolution inclusions of Clarke's subdifferential type in Hilbert spaces. The existence of mild solutions and controllability results are given and proved by using stochastic analysis techniques, semigroup of operators theory, a fixed point theorem of multivalued maps and properties of generalized Clarke subdifferential. An example is included to illustrate the applicability of the main results.


Introduction
It is well known that controllability plays a significant role in the concept of control theory and engineering.Currently, fruitful achievements have been obtained on controllability of stochastic systems and inclusion problems, see e.g.Bashirov and Mahmudov [1], Mahmudov [20], Obukhovski and Zecca [24] and Rykaczewski [27] and the references therein.In addition, the controllability problems for stochastic differential equations have become a field of increasing interest due to its applications in economics, ecology and finance.More precisely, Klamka [6][7][8][9][10] studied stochastic controllability systems with different kind of delays.Lin and Hu [11] considered the existence results of stochastic inclusions with nonlocal initial conditions.Sakthivel et al. [29,30] obtained the approximate controllability of semilinear fractional differential systems in Hilbert spaces.Ren et al. [26] studied the controllability of impulsive neutral stochastic differential inclusions with infinite delay.
Recently, many researchers have paid increasingly attention to the evolution inclusions with Clarke's subdifferential type which have have been studied in many papers, we refer the readers to [12-18, 22, 23, 32, 33] and the references therein.In fact, Clarke's subdifferential has important applications in mechanics and engineering, especially in nonsmooth analysis and optimization [2,23].At present, although some significant results have been obtained for the solvability and control problems of evolution inclusions of generalized Clarke subdifferential, it seems that there are still many interesting ideas and unanswered questions.However, the study of the controllability of the systems described by stochastic evolution inclusions of generalized Clarke subdifferential in Hilbert spaces has not been investigated yet and the investigation on this topic has not been appreciated well enough.
Motivated by the above consideration, we will study the existence of mild solutions and controllability of the following stochastic evolution inclusions of generalized Clarke's subdifferential type with nonlocal initial conditions: dx(t) ∈ (Ax(t) + Bu(t))dt + σ(t, x(t)) dw(t) + ∂F(t, x(t)) dt, t ∈ J = [0, b], x(0) = x 0 + g(x), (1.1) where x(•) takes the value in the separable Hilbert space H, A : •) : H → R; σ and g are given appropriate functions to be specified later; w is a Q-Wiener process on a complete probability space (Ω, Γ, P) and x 0 is Γ 0 measurable H-valued random variable independent of w.If the operator A is monotone, there are a lot of results in this direction (cf.[31]).The rest of this paper is organized as follows.In Section 2, we will recall some useful preliminary facts.In Section 3, the existence of mild solutions of the system (1.1) is established and proved by applying stochastic analysis techniques, semigroup of operators theory, a fixed point theorem of multivalued maps and properties of generalized Clarke subdifferential.In Section 4, the controllability of the system (1.1) is formulated and proved mainly by using a fixed point technique.Finally, an example is given to illustrate our main results in Section 5.

Preliminaries
Let (Ω, Γ, {Γ t , t ≥ 0}, P) be a complete probability space equipped with a normal filtration {Γ t , t ≥ 0} satisfying that Γ 0 contains all P-null sets of Γ. E(•) denotes the expectation of a random variable or the Lebesgue integral with respect to the probability measure P. Let H, U be separable Hilbert spaces and {w(t), t ≥ 0} be a Wiener process with the linear bounded covariance operator Q such that tr Q < ∞.
We assume that there exist a complete orthonormal system {e k } k≥1 in H, a bounded sequence of nonnegative real numbers λ k such that Qe k = λ k e k (k = 1, 2, . . . ) and a sequence of independent Brownian motions {β k } k≥1 such that e ∈ H, t ≥ 0 and Γ t = Γ w t , where Γ w t is the σ-algebra generated by {w(s H, H) be a space of all Hilbert-Schmidt operators from Q For details, we refer the reader to [3,28] and references therein.Next, we introduce some basic definitions on multivalued maps, for more details, please refer to the books [4,5].
For a Banach space X with the norm • , X * denotes its dual and •, • the duality pairing of X and X * .For convenience, we use the following notations: Definition 2.1.Given a Banach space X and a multivalued map G : X → 2 X \ ∅ = P(X), we say (iii) G is upper semicontinuous (u.s.c.) on X if for each x 0 ∈ X, the set G(x 0 ) is a nonempty closed subset of X, and if for each open set U of X containing G(x 0 ), there exists an open neighborhood V of x 0 such that G(V) ⊆ U.
(iv) G is completely continuous if G(B) is relatively compact for every bounded subset B ∈ P(X).
(v) G has a fixed point if there is a x ∈ X such that x ∈ G(x).
Now, recall the definition of the generalized gradient of Clarke for a locally Lipschitzian functional F : X → R. From [2], we denote by F 0 (x; v) the Clarke generalized directional derivative of F at x in the direction v, that is λ and we denote by ∂F, which is a subset of X * given by the generalized gradient of F at x (the Clarke subdifferential).
The following basic properties play important roles in our main results.
(ii) for every x ∈ Ω, the gradient ∂F(x) is a nonempty, convex, weak * -compact subset of X * and x * X * ≤ Λ for any x * ∈ ∂F(x) (where Λ > 0 is the Lipschitz constant of F near x); (iii) the graph of the generalized gradient ∂F is closed in Ω × X * w * topology, i.e., if {x n } ⊂ Ω and {x * n } ⊂ X are sequences such that x * n ∈ ∂F(x n ) and x n → x in X, x * n → x * weakly * in X * , then x * ∈ ∂F(x) (where X * w * denotes the Banach space X * furnished with the w * -topology); (iv) the multifunction Ω x → ∂F(x) ⊆ X * is u.s.c.from Ω into X * w * .Lemma 2.3 (Proposition 3.44 of [23]).Let X be a separable reflexive Banach space, 0 < b < ∞ and h : (0, b) × X → R be a function such that h(•, x) is measurable for all x ∈ X and h(t, •) is locally Lipschitz on X for all t ∈ (0, b).Then the multifunction (0, b) × X (t, x) → ∂h(t, x) ⊂ X * is measurable, where ∂h denotes the Clarke generalized gradient of h(t, •).
At the end of this section, we present the following lemma and fixed point theorem that are the key tools in our main results.
ds for all 0 ≤ t ≤ b and p ≥ 2, where L G is the constant involving p and b.

Theorem 2.7 ([19]
).Let X be a locally convex Banach space and F : X → 2 X be a compact convex valued, u.s.c.multivalued map such that there exists a closed neighborhood V of 0 for which F (V) is a relatively compact set.If the set is bounded, then F has a fixed point.

Existence of mild solutions
In this section, we study the existence of mild solutions for the system (1.1).Firstly, according to the book [25], we may define a mild solution of problem (1.1) as follows.
In the following, we impose the following hypotheses.
(H1) A : D(A) ⊆ H → H is the infinitesimal generator of a C 0 -semigroup T(t)(t ≥ 0) and the semigroup T(t) is compact for t > 0.
By Theorem 1.2.2 of [25], there exist constants ≥ 0 and M ≥ 1 such that (H2) F : J × H → R satisfies the following assumptions: •) is locally Lipschitz continuous for a.e.t ∈ J; (iii) there exist a function a ∈ L 1 (J, R + ) and a constant c ≥ 0 such that for a.e.t ∈ J and all x ∈ H.
(H3) σ : J × H → L 2 0 is continuous in the second variable for a.e.t ∈ J and there exist a function η ∈ L 2 (J, R + ) and a constant d ≥ 0 such that (H4) g : C(J, H) → H is continuous and there exists a constant e ≥ 0 such that g(x) 2 ≤ e(1 + x 2 ).
Next, we define an operator N : To obtain our main results, we also need the following lemmas.
Lemma 3.2.If the assumption (H2) holds, then for each x ∈ L 2 Γ (J, H), the set N (x) has nonempty, convex and weakly compact values.
Proof.The main idea of the proof comes from Lemma 5.3 of [23] and Lemma 2.6 of [16].
Firstly, from Lemma 2.2 (ii), ∂F(t, x) is nonempty, convex and weakly compact in the Hilbert H and ∂F is P wkc (H)-valued.Thus N (x) has convex and weakly compact values.
Next, we will prove that N (x) is nonempty.Let x ∈ L 2 Γ (J, H), then there exists a sequence {ϕ n } ⊆ L 2 Γ (J, H) of simple functions such that Hence, {ζ n } remains in a bounded subset of L 2 Γ (J, H).Thus, we can suppose that Hence by (3.2), we have where the Kuratowski upper limit (cf.Definition 3.14 of [23]) of set ∂F(t, ϕ n (t)) is given by  [22]).If (H2) holds, the operator N satisfies: if x n → x in L 2 Γ (J, H), w n → w weakly in L 2 Γ (J, H) and w n ∈ N(x n ), then we have w ∈ N (x).
Now, we study the existence of mild solutions for the system (1.1).
Theorem 3.4.For each u ∈ L 2 Γ (J, U), if the hypotheses (H1)-(H4) are satisfied, then the system (1.1) has a mild solution on J provided that Proof.Firstly, for any x ∈ C(J, L 2 (Γ, H)) ⊂ L 2 (J, H), from Lemma 3.2, we can consider the multivalued map F : C(J, L 2 (Γ, H)) → 2 C(J,L 2 (Γ,H)) defined by It is clear that problem (1.1) is reduced to find a fixed point of F .We will show that the operator F satisfies all the conditions of Theorem 2.7.Next, to complete the proof, we divide the proof into six steps.
Step 2: The operator F is bounded on bounded subset of C(J, L 2 (Γ, H)).
Similarly, we have Hence, using the compactness of T(t) (t > 0), we conclude that the right-hand side of the above inequalities tends to zero as τ 2 − τ 1 → 0. Thus we conclude F (x)(t) is continuous from the right in (0, b].Similarly, for τ 1 = 0 and 0 < τ 2 ≤ b, we may prove that E ϕ(τ 2 ) − x 0 2 tends to zero independently of x ∈ B r as τ 2 → 0. Hence, by the above arguments, we can deduce that {F (x) : x ∈ B r } is an equicontinuous family of functions in C(J, L 2 (Γ, H)).
Step 4: F is completely continuous.
According to Definition 2.1 (iv), we will show the set Π(t) = {ϕ(t) : ϕ ∈ F (B r )} is relatively compact in H for t ∈ J be fixed.To this end, taking account Steps 2-3 and making use of Ascoli-Arzelà theorem, we have to prove that the set Clearly, Π(0) = {x 0 } is compact.So we consider t > 0. Let 0 < t ≤ b be fixed.For any x ∈ B r , ϕ ∈ F (x), there exists an f ∈ N (x) such that (3.4) holds for each t ∈ J.For each ε ∈ (0, t), t ∈ (0, b] and any x ∈ B r , we define From the boundedness of which implies the set Π(t) (t > 0) is totally bounded.In view of Step 3, it is relatively compact in H, which completes the proof of Step 4.
Step 5: F has a closed graph.Let is bounded.Hence, passing to a subsequence if necessary, It follows from (3.5), (3.6) and the compactness of the operator T(t) that From Lemma 3.3 and (3.7), we obtain f * ∈ N (x * ).Thus we have shown that ϕ * ∈ F (x * ), which implies that F has a closed graph.By Proposition 3.3.12(2) of [23], F is u.s.c.
By Steps 1-5, we have obtained that F is compact convex valued and u.s.c., F (B r ) is a relatively compact set.According to Theorem 2.7, it remains to prove the set Let x ∈ Ω and suppose that there exists a f ∈ N (x) such that Then by the assumptions (H1), (H2) (iii), (H3) and (H4), we obtain where Since K < 1, from (3.8), we obtain Hence, the set Ω is bounded.By Theorem 2.7, F has a fixed point.The proof is completed.

Controllability results
In this section, we mainly investigate the complete controllability of the system (1.1).The following definition of the controllability is standard.We state it here for the sake of convenience.
Definition 4.1 (Complete controllability).The system (1.1) is said to be completely controllable on the interval J if, for every x 0 , x 1 ∈ H, there exists a stochastic control u ∈ L p Γ (J, U)(p > 1) which is adapted to the filtration {Γ t } t≥0 such that a mild solution x of system (1.1) satisfies x(b) = x 1 .
Theorem 4.2.Suppose that the assumptions (H1)-(H5) are satisfied.Then the system (1.1) is completely controllable on J provided that Using the control u α and the assumptions, it is easy to see that the multivalued map F u is well defined and x 1 ∈ (F u x)(b).Thus to obtain the complete controllability, we only need to prove that F u has a fixed point.
The proof is similar to Theorem 3.4.To complete the proof, a simple version of proof is given.
Step 2: The operator F is bounded on bounded subset of C(J, L 2 (Γ, H)). Let In fact, it is enough to show that there exists a positive constant 0 such that for each ϕ ∈ F where u α is given by (4.1).Then notice that , there exists a f ∈ N (x) such that for each t ∈ J, we have ϕ as (4.2).Using the estimation on E u α (t) 2 similarly to Step 3 of Theorem 3.4, we know that {F u (x) : x ∈ B ζ } is equicontinuous family of functions in C(J, L 2 (Γ, H)).
Step 4: F u is completely continuous.
Let t ∈ J be fixed.We show that the set , there exists f ∈ N (x) such that ϕ(t) satisfies (4.2).For each ∈ (0, t), t ∈ (0, b] and any x ∈ B ζ , we can use the way in Theorem 3.4 to prove that the set Taking account Steps 2-3 and making use of Ascoli-Arzelà theorem, we obtain that F u is completely continuous.
Step 5: F u has a closed graph. Let From the compactness of T(t), (4.3) and (4.4), we obtain Note that ϕ n → ϕ * in C(J, L 2 (Γ, H)) and f n ∈ N (x n ).From Lemma 3.3 and (4.5), we obtain f * ∈ N (x * ).Hence, we have proved that ϕ * ∈ F u (x * ), which implies that F u has a closed graph.It follows from Proposition 3.3.12(2) of [23] that F u is u.s.c.
Next, to write the above system (5.1)into the abstract form of (1. u n e n ∈ U. Under the above assumptions, we know that the system (5.1) can be written in the abstract form (1.1) and all the conditions of Theorem 4.2 are satisfied.Therefore, by Theorem 4.2, stochastic control system (5.1) is completely controllable on J = [0, b].

(
H5) The linear operator W : L 2 Γ (J, U) → H, defined by Wu = b 0 T(b − s)Bu(s) ds has an inverse operator W −1 which takes value L 2 Γ (J, H)/ ker W and there exist two positive
[23]over, (H2) (iii) and Lemma 2.2 (iv) imply that x → ∂F(t, x) is u.s.c.Recalling that the graph of an u.s.c.multifunction with closed values is closed (cf.Proposition 3.12 of[23]), we obtain that for a.e.t ∈ J, if which implies that N (x) is nonempty.The proof is completed.