Approximate Controllability of Fully Nonlocal Stochastic Delay Control Problems Driven by Hybrid Noises

In this paper, a class of time-space fractional stochastic delay control problems with fractional noises and Poisson jumps in a bounded domain is considered. The proper function spaces and assumptions are proposed to discuss the existence of mild solutions. In particular, approximate strategy is used to obtain the existence of mild solutions for the problem with linear fractional noises; fixed point theorem is used to achieve the existence of mild solutions for the problem with nonlinear fractional noises. Finally, the approximate controllability of the problems with linear and nonlinear fractional noises is proved by the property of mild solutions.

Time fractional derivatives are used to describe phenomenon of early arrival or to tail in time. Spatial fractional derivatives are more suitable for describing nonlocal and scale effects. There are many applications of fractional derivatives in anomalous diffusion, random walk, nonlocal elasticity, and memory materials, see [1][2][3][4][5][6][7] and the references therein. The fractional derivatives in problem (1) are constant-order fractional derivatives. In recent years, there has been a series of papers studying variable-order fractional derivatives and distributed-order fractional derivatives. These general fractional derivatives are applied to model the complex interaction and superposition systems of nonlocal effects and memory effects on multi-scales, see [8][9][10][11].
When α = 1, γ ∈ (0, 1), H = 1, and h ≡ 0, Wang [12] discussed the following equation in R N : dy(t, x) + (−∆) γ y(t, x) = f (t, x, y τ (t, x)) + g(t, y τ (t, x))dB(t), where B is a Wiener process. He obtained the existence and uniqueness of solutions and weak pullback mean random attractors. Mohammed [13] proved the existence of approximate solutions on a bounded domain to the time-fractional case, i.e., α ∈ (1/2, 1), γ = 1, H = 1 and h ≡ 0. However, we consider the time-space fractional case, which is called the fully nonlocal case. In the last decade, fully nonlocal partial differential equations have attracted great attention. Several different approaches to simulate fully nonlocal PDE were presented in [14][15][16][17][18]. Fundamental solutions for fully nonlocal PDE problems were discussed in [19][20][21]. These results encourage researchers to study the fractional stochastic partial differential equations (f-SPDE). For example, f-SPDE driven by fractional Brownian motions with delay were considered in [22][23][24], and the existence of mild solutions was obtained under suitable assumptions. Caraballo, Garrido-Atienza and Taniguchi [25] considered the following problem driven by linear fractional noise: where A is a bounded abstract operator and X is a Hilbert space. They obtained the existence and uniqueness of mild solutions. Li [23] discussed problem (2) with fractional time derivative, i.e., C 0 D α t y(t) instead of dy(t) in (2) and established the existence and uniqueness of mild solution by approximate method when indexes s ∈ (1/2, 1] and H ∈ (1/2, 1). As for nonlinear fractional noises, Pei and Xu [24] considered the following problem: dy(t) = A(t, y(t))dt + f (t, y τ (t))dt + g(t, y τ (t))dB(t) + U h(t, y τ (t); u)θ(du, dt), t ∈ (0, T], The existence and uniqueness of mild solution were obtained under Lipschitz-type conditions. For more papers, we recommend [26,27]. We are interested in the controllability of problem (1). Controllability is a basic topic in control theory. It has wild applications in the real world, see [28,29] and the references therein. Recently, approximate controllability of f-SPDE was studied by many researchers. In [30], Lakhel discussed a class of integro-differential equations with linear fractional noises, By means of resolvent operators, controllability of (3) was obtained when Hurst index H ∈ (1/2, 1). In [31], Ahmed considered the following equation driven by nonlinear fractional noises and Poisson jumps, where α ∈ (1, 2) and H ∈ (1/2, 1). Ahmed proved the approximate controllability under a uniform boundness condition For more papers, we recommend [32,33]. Note that (4) is a common condition to discuss the approximate controllability of control problems, see, for examples, refs [34][35][36]. However, in this paper, we obtain the approximate controllability by the property of mild solutions.
Problem (1) strongly depends on the ranges of α, γ, H, and N. In this paper, we suppose 1/2 < α, γ, H < 1, and N > 2γ. Compared with the aforementioned papers, there are two main contributions in this paper:

1.
Utilize a reasonable framework of mild solutions and overcome the complex calculations caused by not only fractional differential operators but also fractional Brownian motions and Poisson jumps. Establish the existence and uniqueness of the mild solution to fully nonlocal stochastic delay control problems with both linear and nonlinear fractional noise by two different methods.

2.
Establish sufficient conditions to the approximate controllability of fully nonlocal stochastic delay control problems. The approximate controllability result is based on the controllability theory of deterministic control problems.
The paper is organized as follows. In Section 2, we introduce basic concepts and results. In Section 3, we prove the existence and uniqueness of mild solutions of problem (1) for linear fractional noises and nonlinear fractional noises. In Section 4, we study approximate controllability. In Section 5, we give conclusions.

Preliminaries
In this section, we introduce basic concepts and results.

Fractional Brownian Motions
Let (Σ, F , P, F t ) be a complete probability space with a filtration {F t } t∈[0,∞) which satisfies usual conditions. A one-dimensional fractional Brownian motion {β H (t)} t∈[0,T] is a centered Gaussian process which has zero mean and its covariance is Fractional Brownian motions can be expressed by Wiener processes, see [37]. Let W be a real separable Hilbert space, {w i } ∞ i=1 be an orthonormal basis of W and L (W) be the space of all bounded linear operators on W. Let A ∈ L (W) be a symmetric, nonnegative operator and is a sequence of independent one-dimensional fractional Brownian motions. B H A has zero mean and covariance Cov B H A (t), µ B H A (τ), ν = Cov(t, τ) Aµ, ν , for any t, τ ∈ [0, T] and µ, ν ∈ W. In this paper, we consider the cylindrical fractional Brownian motion B H , i.e., λ i = 1, i = 1, 2, · · · . Let L 0 2 (W, L 2 (O)) be the space of Hilbert-Schmidt operators. An operatorḡ ∈ L 0 2 (W, L 2 (O)) satisfiesḡ ∈ L (W, L 2 (O)) and Endowed the inner product ḡ,g L 0 ) is a separable Hilbert space. We recall the following inequality:

Poisson Jumps
Let (U, B(U), e) be a σ-finite measurable space and {π t } t∈[0,∞) be a stationary Poisson point process which is defined on (Σ, F , P) and take values in U. The compensated Poisson martingale measureθ is defined bỹ where θ is a counting measure which is generated by {π t } t∈[0,∞) .θ has zero mean and variance E[θ(du, t)] 2 = te(du).
Let M e,2 F ([0, T] × U, L 2 (O)) be the space of F × B(U) measurable processes with finite second moments and be equipped with the norm is a L 2 (Σ; L 2 (O))-valued random process which has zero mean. Furthermore, recall Theorem 6.1 in [39]: holds. Moreover, the following inequality holds

Fractional Differential Operators
Consider the deterministic problem without time delay The Riemann-Liouville kernel function is given by R α (t) = t α−1 Γ(α) , t, α > 0, where Γ denotes the Gamma function. Denote Laplace transform by L, then where denotes convolution. The αth Riemann-Liouville fractional derivative and Caputo fractional derivative are defined by respectively. The connection between R D α t and c 0 D α t is given by for sufficiently smooth function f . There are several approaches to define the fractional Laplacian. We introduce the definition by means of Fourier transform F , Let the space We recall the embedding result in [40].
The Mittag-Leffler function E k 1 ,k 2 and the Mainardi function M k are defined by , for all t > 0. By Laplace transform in time variable and Fourier transform in spatial variable on both sides of the equation in (5), the formal solution of problem (5) is given by Furthermore, denote Then, equivalent to (6), the formal solution y satisfies Recall the properties of Y α γ , Z α γ , η and ξ. Let Let

Function Spaces
Let L 2 (Σ; L 2 (O)) be the set of all F measurable random variables ζ such that We use notation L 2 F t ([−τ, T]; H γ 0 (O)) to denote the space of all F t -adapted random processes such that ) be the space of all continuous random processes from [−τ, T] to L 2 (O), a.s., with essentially finite second moments and be endowed with the following norm y 2 Let D (T) be the set of all stochastic processes with the following properties: Then the definition of mild solutions is given by Definition 1. We call y ∈ D (T) a mild solution of problem (1) if for any t ∈ [0, T], y satisfies the following formula or equivalently, if each integral is well defined.

Remark 1.
Similar to the proof in [41], we can prove that a classic solution is also a mild solution defined in the sense of Definition 1.

Mild Solutions
In this section, we prove the existence and uniqueness of the mild solution of problem (1) with linear fractional noise and nonlinear fractional noise by approximate method and the Banach Fixed Point Theorem.
We propose the following hypotheses on nonlinear functions f and h: where constants M 1 , L 1 > 0.
Hypothesis 2 (H2). For any t ∈ [0, T] and x ∈ O, there exist a positive function Φ ∈ L 1 (O) and a constant M 2 > 0 such that

Linear Fractional Noises
In this subsection, we discuss the following problem: Assume that The main result of this subsection is as follows. Proof. We use the approximate strategy to prove the result. The proof is divided into four parts. First, we consider the regularity of stochastic integrals. Second, we construct a sequence {y n } ∞ n=1 to approximate the solution of problem (9). Third, we show that the limit of {y n } ∞ n=1 in D (T) is a mild solution of problem (9). At last, we prove the uniqueness of mild solutions by the Gronwall inequality.
By similar arguments to J 4 and J 2 , for any λ > 0, we obtain Thus, we summarize that Step 2: Approximate solutions. Let with y n (t, x) = ϕ(t, x), a.s., for all t ∈ [−τ, 0] and x ∈ O, n = 1, 2, · · · . Next, we show that {y n } ∞ n=1 is a sequence in D(T). It is clear that y 1 (t, x) ∈ D(T) by Lemma 5. Let y n , n ≥ 2, belong to D(T). We show that y n+1 belongs to D(T). Since we have already proved the regularity of K 3 (t, x) and K 4 (t, x) in Step 1, then we remain to prove that K 1 (t, x) and K 2 (t, x) belong to D(T).
Step 3: Existence of solutions. First, we show that {y n } ∞ n=1 is a convergent sequence in D(T). Similar discussions to J 4 and J 5 , by (H1) and (H2), we obtain Let Ψ n (t) = sup E y n+1 (s, ·) − y n (s, ·) 2 2 , then we get for any t ∈ [0, T]. Moreover, Ψ n satisfies We assert that y is a mild solution of problem (9). Consider as n → ∞. The assertion is proved, i.e., y is a mild solution to the problem (9).
Proof. First, we show that there exists a solution mapping. By (H4), we deduce that for any t ∈ [0, T] and φ ∈ D (T), Thus, by Theorem 1, there exists a unique mild solution y of the following problem: Next, we demonstrate that Ψ is a contraction map. For anyφ,φ ∈ D (T), consider Further, similar to the proof of the uniqueness of mild solution to the problem (9), we get in review of (H4). We conclude that , T}, then Ψ is a contraction mapping on D (T 1 ). Furthermore, by the Banach Fixed Point Theorem, there exists a unique mild solution of problem (1) on [−τ, T 1 ]. By taking T 1 as the initial time and repeating this process, we can extend the mild solution to [−τ, T] in finite steps.

An Estimate
According to the proof of Theorems 1 and 2, we observed that mild solutions are bounded in D(T). In particular, we get the following results. Proposition 2. Let 1/2 < α, γ, H < 1, N > 2γ, (H1), (H2) and (H3) hold. If y is a mild solution of problem (9), then there exists a positive constant M such that y 2 Proof. Since y(t, x) = ϕ(t, x), a.s., for all t ∈ [−τ, 0] and x ∈ O, we obtain that On the other hand, for any t ∈ (0, T], we get First, we consider Second, we get By similar discussions to (10), we obtain Further, Therefore, we conclude that for any t ∈ [−τ, T], where C(·) depends on the parameters in brackets. By taking supremum on both sides and the Granwall inequality, we get that there exists M > 0 such that Then, by (H4) we get that Hence, by a similar proof to Proposition 2, we obtain

Approximate Controllability
In this section, we apply the results in Section 3 to prove the approximate controllability of problem (9) and problem (1).
To illustrate approximate controllability of control problem (9) and problem (1), we recall the following result.

Lemma 6 ([34]
). For any ρ ∈ L 2 (Σ; L 2 (O)), there exists ζ such that Proof. It is necessary to prove that there exists a sequence of controls whose limit transfers initial state ϕ to y(T) at time T. The construction of the sequence is inspired by [42].
We will show that {v κ } ∞ κ=1 is the sequence that we needed.

Conclusions
In this paper, we discuss a class of stochastic control problems with Caputo fractional derivatives, fractional Laplacian operators, fractional Brownian motions and Poisson jumps. We consider the problems with linear and nonlinear fractional noises by different methods. We obtain the existence and uniqueness of mild solutions when 1/2 < α, γ, H < 1, and N > 2γ. Furthermore, we apply the results obtained for mild solutions to get the approximate controllability of the stochastic control problems.