Controllability of fractional stochastic evolution equations with nonlocal conditions and noncompact semigroups

Abstract This article deals with the exact controllability for a class of fractional stochastic evolution equations with nonlocal initial conditions in a Hilbert space under the assumption that the semigroup generated by the linear part is noncompact. Our main results are obtained by utilizing stochastic analysis technique, measure of noncompactness and the Mönch fixed point theorem. In the end, an example is presented to illustrate that our theorems guarantee the effectiveness of controllability results in the infinite dimensional spaces.


Introduction
In this article, we shall be concerned with the controllability for the following fractional evolution equations with nonlocal initial conditions of the form: where D t α c is the Caputo fractional derivatives of order < < α 1 1 2 . Let and be two separable Hilbert spaces and the state (⋅) x takes its values in .

( ) ⊂ → A D A
: is a closed linear operator, and −A is the infinitesimal generator of a C 0 -semigroup ( )( ≥ ) T t t 0 on . For convenience, we will use the same notation ∥⋅∥ to denote the norms in and and use (⋅ ⋅) , to denote the inner product of and without any confusion. We are also employing the same notation ∥⋅∥ for the norm of ( ) L , , which denotes the space of all bounded linear operators from to . Suppose that { ( ) ≥ } W t t : 0 is a -valued Brownian motion or Wiener process with a finite trace nuclear covariance operator ≥ Q 0 defined on a filtered complete probability space , . The control function (⋅) u belongs to the space ( ) L J, 2 , a Banach space of admissible control functions, for a separable Hilbert space , → B: is a bounded linear operator. f σ , and g are appropriate functions to be given later. x 0 is an 0 measurable -value random variable.
In recent years, Stochastic differential equations have attracted great interest due to their successful applications to problems in mechanics, electricity, economics, physics and several fields in engineering. For details, see [8,9,[32][33][34][35][36][37][38] and references therein. In particular, some researchers investigated controllability of stochastic dynamical control systems in infinite dimensional spaces [25][26][27][28][29]. It is generally known that nonlocal initial conditions can be applied in physics with better effect than the classical Cauchy problem traditional initial condition. However, to the best of our knowledge, the exact controllability of stochastic control systems with nonlocal conditions of the form (1.1) has not yet been studied. Therefore, a natural problem is as follows: how to investigate the exact controllability of stochastic evolution equations involving noncompact semigroups?
Motivated by this consideration, in this article we establish the exact controllability for fractional stochastic evolution equations with nonlocal conditions of the form (1.1) in a Hilbert space under the assumption that the semigroup is noncompact. By using some constructive control functions, we transfer the controllability problem into a fixed-point problem and then apply the measure of noncompactness and the Mönch fixed point theorem to discuss the controllability for problem (1.1). We delete the compactness of semigroup ( )( > ) T t t 0 , so our theorems guarantee the effectiveness of controllability results in the infinite dimensional spaces. Furthermore, we give a useful way to discuss stochastic control systems with noncompact semigroups.
We organize the article in the following way. In Section 2, we introduce some useful definitions and preliminary results to be used in this article. In Section 3, we state and prove the exact controllability results for fractional stochastic evolution equations with nonlocal conditions. Finally, in Section 4, an example is provided to illustrate the applications of the obtained results.

Preliminaries
In this section, we introduce notations, definitions and preliminary facts, which are used throughout this article. Let be a filtered complete probability space satisfying the usual condition, which means that the filtration is a right continuous increasing family and 0 contains all P-null sets. , where denotes the expectation with respect to the measure P. Let ( ( )) C J L Ω , , 2 be the Banach space of all continuous mappings from J to . We use to denote the space of all t -adapted measurable processes ∈ ( ( )) x C J L Ω , , 2 endowed with the norm ∥ ∥ = ( ∥ ( )∥ ) ∈ / x xt sup t J 2 1 2 . The theory of stochastic integrals in a Hilbert space can be found in [34,36].
In the rest of the manuscript, we suppose that A generates an equicontinuous C 0 -semigroup ( )( ≥ ) T t t 0 of uniformly bounded linear operator in . That is, there exists a positive constant ≥ . Evidently, B r is a bounded closed convex set in .
By [29, Proposition 2.8], we have the following result which will be used throughout this article. , The Riemann-Liouville fractional integral of order > α 0 of a function ( +∞) → y: 0, is given by provided that the right side is pointwise defined on ( +∞) 0, .
The Caputo fractional derivative of order > α 0 of a function [ +∞) → y: 0, is given by For ∈ x , we define two operators ( )( ≥ ) t t 0 α and ( )( ≥ ) t t 0 α as follows: and ( ) t α are linear and bounded operators in , i.e., for any ∈ x , Now, we recall some properties of the measure of noncompactness, which will be used later.
Let (⋅) μ denote the Kuratowski measure of noncompactness of the bounded set. For the details of the definition and properties of the measure of noncompactness [39]. Let X be a Banach space, for any The following lemmas are needed in our arguments.
n be a bounded and countable set. Then, ( ( )) μ D t is the Lebesgue integral on J, and In this article, we adopt the following definition of the mild solution of (1.1) based on [27,36].
x t is measurable and adapted to t ; (ii) ( ) x t satisfies the following integral equation: be the state value of system (1.1) at terminal time b corresponding to control u. The set is called the reachable set of (1.1) at the terminal time b.
To prove the controllability result, the following hypotheses are necessary throughout the article: , and there exist two for any countable subset ⊂ D .
(H2) The function × → f J : satisfies: (i) For any ∈ t J, ( ⋅) f t, is continuous, and (⋅ ) f x , is strongly measurable for all ∈ x ; (ii) For some > r 0, there exists a function ∈ ( ) such that , . such that is continuous and there exists a constant > K 0 such that for any > r 0 , . For any ∈ ( ) x L Ω, 1 2 , we introduce a control ( ) ≔ ( Next, we estimate some properties of control ( ) u t defined above.
Lemma 2.5. Suppose that assumptions (H1)-(H4) are satisfied, then for any ∈ x B r , the following conclusions hold: Moreover, for all ∈ t J, using the Hölder inequality and Lebesgue dominated convergence theorem, we can get  At the end of this section, we present the Mönch fixed point theorem, which plays a key role in our proof of controllability for system (1.1).
Lemma 2.6. [42] Let Ω be a closed convex subset of a Banach space X and ∈ θ Ω. Assume that → P Ω Ω : is a continuous map, which satisfies Mönch's condition, i.e., for ⊂ D Ω is countable and ⊂ ({ } ∪ ( )) ⇒ D co θ P D D is compact. Then, P has at least one fixed point in X.

Main results
In this section, we shall discuss the exact controllability of the fractional stochastic dynamical control system (1.1) by using the measure of noncompactness and Mönch fixed point theorem. and Proof. To begin with, we define an operator → F: as follows: It is easy to see that the mild solution of control system (1.1) is equivalent to the fixed point of the operator F defined by (3.3). Next, we complete the proof by four steps.
Step   Dividing both sides by r and taking the lower limit as → +∞ r , we obtain   Step 3. We prove that ( ) F B r is an equicontinuous family of functions on [ ] b 0, . For any ∈ x B r and ≤ < ≤ t t b 0 1 2 , we get that  I I  I  I  I  I  I   7  7 , ,d  From assumption (3.2), we know that < ρ 1, which means ( ) = μ D 0, that is, D is relatively compact. By Lemma 2.6, F has at least one fixed point ∈ x B r , which is a mild solution of the fractional stochastic control system (1.1) and it satisfies ( ) = x b x 1 for any ∈ ( ) x L Ω, 1 2 . Therefore, the fractional stochastic control system (1.1) is exact controllable on J.
This completes the proof of Theorem 3.1. □ Remark 3.2. Impulsive effects exist in many evolution processes, so it is necessary to study impulsive stochastic evolution equations. Upon making some appropriate assumptions, by applying the ideas and techniques as in this article, one can obtain the exact controllability results for a class of fractional impulsive stochastic evolution equations with nonlocal conditions.