Nonlocal Controllability of Sobolev-Type Conformable Fractional Stochastic Evolution Inclusions with Clarke Subdifferential

In this paper, Sobolev-type conformable fractional stochastic evolution inclusions with Clarke subdifferential and nonlocal conditions are studied. By using fractional calculus, stochastic analysis, properties of Clarke subdifferential and nonsmooth analysis, sufficient conditions for nonlocal controllability for the considered problem are established. Finally, an example is given to illustrate the obtained results.

because fractional derivatives have been proved that they are a very good way to model many phenomena with memory in various fields of science and engineering [14][15][16][17][18]. Khalil et al. [19] introduced a novel definition named conformable fractional derivative which is an extension of the classical limit definition of the derivative and obeys the classical properties including linearity property, product rule, quotient rule, Rolle's theorem and mean value theorem and coincides with the classical definition of Riemann-Liouville and Caputo on polynomials up to a constant multiple.
In recent years, there have been a lot of results on controllability problems with Clarke subdifferential. Li and Lu [20] discussed the existence and controllability for stochastic evolution inclusions of Clarke's subdifferential type. Zhenhai et al. [21] studied optimal feedback control and controllability for hyperbolic evolution inclusions of Clarke's subdifferential type. Ahmed et al. [22] established sufficient condition for controllability and constrained controllability for nonlocal Hilfer fractional differential systems with Clarke's subdifferential. Raja et al. [23] obtained discussed existence and controllability results for fractional evolution inclusions of order 1 < r < 2 with Clarke's subdifferential type. Zhenhai and Zeng [24] proved existence and controllability for fractional evolution inclusions of Clarke's subdifferential type.
To the best of our knowledge, nonlocal controllability of nonlocal Sobolev-type conformable fractional stochastic evolution inclusion with Clarke subdifferential has not been studied in this connection and this fact is the motivation of the our work.
Consider the Sobolev-type conformable fractional stochastic evolution inclusion with Clarke subdifferential nonlocal conditions in the following form where D ϑ is the conformable fractional derivative, 1 2 < ϑ < 1, A and Q are linear operators on a Hilbert space Z with inner product ·, · and norm · . Let be another separable Hilbert space with inner product ·, · and norm · . Assume {ω(t)} t≥0 is -valued Wiener process with a finite trace nuclear covariance operator ≥ 0. Also, . for L( , Z ), where L( , Z ) denotes the space of all bounded linear operators from into Z . ∂G is the Clarke's subdifferential of G(t, z(t)). The state z(·) takes values in the Hilbert space Z and the control function y(·) is given in L 2 (I , Y ), the Hilbert space of admissible control functions with Y a Hilbert space and the symbol B stands for a bounded linear operator from Y into Z . The mappings m : I × Z → Z , σ : I × Z → 2 Z , nonempty, bounded, closed and convex (BCC) multi-valued map, and : I × Z → L ( , Z ) are nonlinear functions and : C(I , Z ) → Z . Here L ( , Z ) denote the space of all -Hilbert-Schmidt operators from to Z .
The main contributions of this paper are summarized as follows: • For the first time in the literature, nonlocal controllability of nonlocal Sobolev-type conformable fractional stochastic evolution inclusion with Clarke subdifferential has been investigated.
• By using fractional calculus, stochastic analysis, properties of Clarke subdifferential, nonsmooth analysis and fixed point theorem, new sufficient conditions for nonlocal controllability of the considered system are derived. • Finally, an example is given to illustrate the theoretical results.

Preliminaries
Definition 2.1 (See [19]) Let 0 < ϑ < 1. The conformable fractional derivative of order ϑ of a function f (·) for t > 0 is defined as follows For t = 0, we adopt the following definition: The fractional integral I ϑ (·) associated with the conformable fractional derivative is defined by Let ( , ϒ, {ϒ t } t≥0 , P) be a complete probability space with a normal filtration {ϒ t } t≥0 satisfying that ϒ 0 contains all P-null sets of ϒ.
Through this paper, letB := C(I , L 2 (ϒ, Z )) be the Banach space of all continuous functions z from I into L 2 (ϒ, Z ), equipped with the supremum norm z B = sup t∈I (E z(t) 2 ) 1/2 , where L 2 (ϒ, Z ) = L 2 ( , ϒ, P, Z ) denotes a Hilbert space of strongly ϒ-measurable, H -valued random variables satisfying E z 2 < ∞. L 2 ϒ (I , Z ) will denote the Hilbert space of all random processes ϒ t -adapted measurable defined on J with values in Z with the norm z L 2 satisfy the following hypotheses (see [25]): Here, (H 1) and (H 2) together with the closed graph theorem imply the boundedness of the linear operator AQ −1 : Z → Z . (H 4) For each t ∈ I and for λ ∈ (ρ(AQ −1 )), the resolvent of AQ −1 , the resolvent R(λ, AQ −1 ) is compact operator.

Lemma 2.1 (See [25]) Let S(t) be a uniformly continuous semigroup. If the resolvent set R(λ, A) of A is compact for every λ ∈ ρ(A), then S(t) is a compact semigroup.
From the above fact, AQ −1 generates a compact semigroup {N (t), t ≥ 0} in Z , which means that there exists T > 1 such that sup t∈I N (t) ≤ T . [26,27]) Let X be a Banach space with the dual space X * and G :

Definition 2.2 (See
, is a subset of X * given by To establish the results, we need the following hypotheses. (H5) There exist positive constants C 1 , (H 6) The function G : I × Z → R satisfies the following conditions: for all z ∈ Z a.e. t ∈ I and z ∈ Z .
(H 7) The function : C(I , Z ) → Z satisfies the following two conditions: (i) there exist positive constants C 5 and C 6 such that E (z) 2 Lemma 2.2 (See [29]) If (A6) holds, then for each z ∈ L 2 ϒ (I , Z ), the set F(z) has nonempty, convex and weakly compact values.

Nonlocal Controllability
Definition 3. 1 The system (1.1) is said to be nonlocal controllable on the interval I , if for every initial condition z 0 and z 1 ∈ Z , there exists a stochastic control y ∈ L 2 (I , Y ) such that a mild solution z(·) of system (1.1) satisfies z(q) + (z) = z 1 , where z 1 and q are the preassigned terminal state and time, respectively. In order to prove the main result, we assume the following hypothesis: (H8) The linear operator U from L 2 (I , Y ) into Z defined by is invertible with inverse operator U −1 taking values in L 2 (I , Y ) \ ker U , and there exist positive constants T 1 and T 2 such that B 2 ≤ T 1 and U −1 2 ≤ T 2 .

Theorem 3.1 If (H1)-(H8) are fulfilled, then (1.1) is nonlocal controllable on I provided that
Proof We define the control operator by using the hypothesis (A8) Consider the map M :B → 2B as follows We will show the operator M has a fixed point. We subdivide the proof into a sequence of steps.
Step 1: For each z ∈B, M (z) is nonempty, convex and weakly compact values.
According to Lemma 2.2, it is easy to see that M (z) has nonempty and weakly compact values. Moreover, as F(z) has convex values, so that if ρ 1 , ρ 2 ∈ F(z) then aρ 1 + (1 − a)ρ 2 ∈ F(z) for all a ∈ (0, 1), which implies clearly that M (z) is convex.
Step 2: The operator M is bounded on bounded subset ofB. Obviously, D ι is a bounded, closed and convex set ofB. We show that there exists a positive constant τ such that for each V ∈ M (z), z ∈ D ι , one has E V (t) 2 ≤ τ.
If χ ∈ M (z), then there exists a ζ ∈ F(z) such that

Thus, M (D ι ) is bounded inB.
Step 3: From the compactness of N (t)(t > 0), we see that the right-hand side of the above inequality tends to zero as t 2 → t 1 . Thus, we can conclude that M (z)(t) is continuous from the right in (0, q]. Similarly, for t 1 = 0 and 0 < t 2 ≤ q, we can prove that E χ(t 2 ) − χ(0) 2 tends to zero uniformly with respect to z ∈ D ι as t 2 → 0. Hence, we infer that {M (z)(t) : z ∈ D ι } is an equicontinuous family of functions inB.
Step 4: M is completely continuous.
We prove that for all t ∈ I , ι > 0, the set (t) = {χ(t) : χ ∈ M (D ι )} is relatively compact in Z . Obviously, (0) is relatively compact in D ι . Let 0 < t ≤ q be fixed, 0 < < t, for z ∈ D ι , we define 2 We see that, when → 0 + , the inequality above tends to zero. Therefore, the set (t) is relatively compact in Z . Thus, from Step 3 and the Arzela-Ascoli theorem, M is completely continuous.
Since ℘ 2 < 1, from (3.7), we obtain Then, z 2B ≤ ℘ 1 1−℘ 2 implies that the set is bounded. By theorem 2.1, M has a fixed point. Therefore, the system (1.1) is nonlocal controllable on I , and the proof is completed.

Example
In this section, we present an example to illustrate the applicability of our results.
Then A and Q can be written as It is well known that AQ −1 generates a strongly continuous semigroup {N (t)} t≥0 which is compact, analytic and self-adjoint in Z . Therefore, with the above choice, the system (4.1) can be written to the abstract (1.1) and all conditions of Theorem 3.1 are satisfied. Thus by Theorem 3.1, the system (4.1) is nonlocal controllable on (0, 1].