Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems

Abstract The aim of this paper is to investigate a class of nonlinear stochastic reaction-diffusion systems involving fractional Laplacian in a bounded domain. First, the existence and uniqueness of weak solutions are proved by using Galërkin’s method. Second, the existence of optimal controls for the corresponding stochastic optimal control problem is obtained. Finally, several examples are provided to demonstrate the theoretical results.


Introduction
In this paper, we discuss a class of stochastic fractional reaction-diffusion systems in Ω: where Ω is a smooth bounded domain contained in d , ∈ ( ) α 0, 1 , ∈ ( +∞) T 0, , u is a vector-valued function, σ is an operator-valued function and { ( )} ∈[ ] W t t T 0, is the space-time noise. Different from Laplacian, the fractional Laplacian is a nonlocal linear operator. A natural question arises: Whether we can extend the results of Laplacian problems to the fractional Laplacian ones or not? Unfortunately, these extensions are not always true, see [1][2][3]. In particular, Devillanova and Carlo Marano have studied the following fractional differential equation in [3]: Devillanova and Carlo Marano have delicately compared the nonfractional cases ( = β 0, 1) and the fractional case ( < < β 0 1) through the mathematical analysis and experimental data. Furthermore, the authors have dealt with the fractional derivative by using the Laplace transform and inversion procedure and some rich results have been obtained. In the last decade, there have been many significant investigations on fractional Laplacian problems in deterministic cases, see [4][5][6] and references therein. But the corresponding stochastic cases need to be further studied. It is well known that (− ) → (− ) Δ Δ α as → α 1. When = α 1, system (1) becomes the standard stochastic reaction-diffusion system. In [7], Ahmed studied on the situation of = α 1 and ≡ g 0, and the well-posedness of weak solution is obtained for ∈ [ +∞) p 1, . We extend this result to the fractional Laplacian case and obtain the existence and uniqueness of weak solutions when ∈ [ / ) * p 1, 2 2 α . Recently, there have been several papers on stochastic fractional system. In [8], Durga and Muthukumar have considered a type of stochastic time-fractional system as follows: where D t α is the αth Caputo type of fractional derivative. Based on fractional calculus, the existence and regularity of mild solutions are obtained, whereas the existence of optimal controls for the corresponding optimal problem is proved by Balder's method. In [9], Bezdek has considered the stochastic fractional reaction-diffusion system on a circle where is a general fractional derivative operator and σ is a locally Lipschitz continuous mapping. Based on the assumption of the growth condition Bezdek proved the existence of mild solutions. In [10], Wang investigated the following stochastic fractional reaction-diffusion system: where φ is the real valued function and ∈ ( +∞) c 0, . Under suitable assumptions on f, the existence and uniqueness of solutions are achieved. Wang et al. [11] keep discussing system (2) on the whole d . But only the existence of solutions is obtained.
Motivated by the aforementioned results, we focus our attention on the properties of weak solutions of system (1). These solutions satisfy (1) in the weak sense with probability 1. Additionally, we wish to extend the optimal control theory of deterministic linear control problems in [12,13] to stochastic control problems. We consider the stochastic control problem of (1) as follows: , is a stochastic control and is a given operator which is called a controller. The framework of this paper is organized as follows. In Section 2, the proper functional spaces and basic concepts are presented. In Section 3, the assumptions on the nonlinear term and noise are stated, then the existence and uniqueness of weak solutions are obtained by using Galërkin's method along with a uniform estimate. In Section 4, the existence of optimal controls is proved. In Section 5, some sufficient conditions and examples are shown to illustrate our results.

Preliminaries
Stochastic fractional reaction-diffusion system (1) involves both stochastic term and fractional Laplacian. It is necessary to employ the theories of stochastic partial differential equations (see [14,15]) and fractional calculus (see [16,17]) to study this type of system.
Let be the Schwartz space, the fractional Laplacian for ∈ φ is defined by Since fractional Laplacian is a nonlocal operator, let : 0 a.e. on \Ω, 1,2, , .
Noting that, for a given function ∈ ( ) f L Ω, p N , it is only defined on Ω, we say ∈ f L p means that we extend the domain of f to d by setting : 0 a.e. in \Ω, 1,2, , It is easily seen that ( ∥⋅∥ ) X , X 0 0 is a Hilbert space under the inner product (1) W is a Gaussian process on Y and (2) W has the following expression: is an adapted random process : ess sup and endow u with the following norm: L T X 0, , , . Moreover, it is a reflexive Banach space for any ∈ [ +∞) p 1, . In this paper, we confine X to be L 2 , X 0 or L p 2 .
We denote by the space of all 0 measurable random variables with bounded second moments. Let We give the concept of weak solutions for system (1) as follows.

The existence and uniqueness of weak solutions
From [18, proposition 9] and [19, proposition 4], we know that there exists a sequence of eigenfunctions of fractional Laplacian which constructs an orthonormal basis of L 2 and an orthogonal basis of X 0 . This result ensures that we can use Galërkin's method to obtain the existence of weak solutions of (1).
First, due to the theory of semigroups of nonlinear operators (see [20,Chapter 4]), we assume that is m-dissipative. We also impose the following conditions on f and the operator-valued function ( ) σ t .
Weak solutions and optimal controls of stochastic fractional systems  1139 We need to show that for any ∈ In virtue of (H2) and basic inequality ( where a 1 and a 2 are constants dependent on { } c c p , , 3 4 . This inequality implies that In addition, by Hölder inequality and (7), we get , , On the other hand, from assumption (H3), we can deduce that for ∀ ∈ ψ L p 2 , ,˜d 0, a.s. Furthermore, since f is lower semi-continuous, choosing = + ψ u ϵμ for any ∈ ( +∞) ϵ 0, The main result of this section is given as follows. Proof. We start with the uniqueness of weak solutions and then we prove the existence of weak solutions by using the Galërkin method. This proof is divided into four steps.
Step 1. The uniqueness of weak solutions. Due to assumption (H3), (3), (4) and Young's inequality, we get that for arbitrarily > ε 0, , then taking the expectation we obtain 1,2, , . Next, we study the following n dimensional stochastic system: Weak solutions and optimal controls of stochastic fractional systems  1141 Because f is m-dissipative, F is m-dissipative. Hence, we can use the linear interpolation methods and Crandall-Liggett's theory to show the existence of solutions of system (10).
is given by is a solution of (10). Thus, = ∑ = u e ϑn j n j j 1 is a solution of (9).
Step 3. A uniform estimate. Integrating by parts on both sides of (9), we obtain Gronwall inequality, we get that . Hence using (11) once more, we deduce that { } u n is bounded in Step 4. The existence of weak solutions for (1). We assert that ũ is a weak solution of (1), that is, we need to show ũ satisfies formula (5). Taking any ∈ ([ ]) ϕ C T 0, where = … i n 1, 2, , . Following the fact that ϕē i is the deterministic function, we will use (12) and the Dominated Convergence Theorem to prove the assertion.
Start with = { ( ( ) ( ) )} I ξ u ϕ e E 0 , 0n i 1 . Noting that ( ) u 0 n is the n-dimensional truncation of u 0 , thus we derive . By the Hölder inequality, there exists Combining (12) with the Dominated Convergence Theorem, we get ,̇¯d E˜,̇¯d , . Therefore, there exists Thus, we conclude that We consider the fourth term I 4 . Since { } u n is bounded in We further consider As g n is the n-dimensional truncation of g, it is clear that Using the Dominated Convergence Theorem, we get , the fact that W n is the n-dimensional truncation of W leads to In other words, ũ satisfies formula (5). □

The existence of optimal controls
Let Ũ be a real Hilbert space, = ∞ L T U 0, ,Ũ be a control functional space and be a bounded linear operator from Ũ to L 2 . We consider the following stochastic control problem: . We consider the cost function with the following form: where ∈ ( ([ ] ) ) L T X L 0, , ; 2 0 2 L and ∈ ( ) U Ũ ,L , the operator satisfies , , where ∈ [ +∞) k 0, being a constant. Let ⊂ U ad U be an admissible set. We call ∈ v U 0 a d be the optimal control of ( ) Thus, we have the following result. Proof. Since U ad is compact, it suffices to prove that Ψ is continuous and J is lower semicontinuous. Let denote the corresponding weak solutions of problem (13) with the same initial state u 0 and g. The proof is divided into two steps.
Step 1. Ψ is continuous. Since { } u k and ū are weak solutions of (13), in the weak sense. From integrating by parts, (H3) and Hölder inequality, we get On the other hand, since weak solutions satisfy inequality (11) This proves that Ψ is continuous.
Step 2. J is lower semi-continuous.
Thus, we get by the arbitrariness of ϵ.
Finally, according to ue to the Fatou lemma, we only need to show , a . s . , inequality (16) holds. Thus, we get that To sum up, J attains its minimum at v. □

Examples
In this section, we give some sufficient conditions and two examples to illustrate the results. Assume that that is, f satisfies (H3). So we deduce the following result. ∈ ( ) U L , 2 L is an identity, is an injection from X 0 into L 2 and is a null operator. Therefore, using Corollary 5.1 and Remark 5.2, we obtain that optimal control problem (17) has at least one optimal control. Weak solutions and optimal controls of stochastic fractional systems  1147 . So we get the existence of optimal controls for optimal control problem (18) by Corollary 5.1 and Remark 5.2.