Fractional neutral stochastic integrodi ﬀ erential equations with Caputo fractional derivative: Rosenblatt process, Poisson jumps and Optimal control

The objective of this paper is to investigate the existence of mild solutions and optimal controls for a class of fractional neutral stochastic integrodi ﬀ erential equations driven by Rosenblatt process and Poisson jumps in Hilbert spaces. First we establish a new set of su ﬃ cient conditions for the existence of mild solutions of the aforementioned fractional systems by using the successive approximation approach. The results are formulated and proved by using the fractional calculus, solution operator and stochastic analysis techniques. The existence of optimal control pairs of system governed by fractional neutral stochastic di ﬀ erential equations driven by Rosenblatt process and pois-son jumps is also been presented. An example is provided to illustrate the theory.


Introduction
Fractional differential equations (FDEs) is about to generalization of the integer order and derivative to arbitrary order. The potential applications of FDEs are in many fields of science and including fluid flow, electrical networks and control theory, see [20,21,22,11,1,2,3,4]. It is well known that many real world problems in science and engineering are modeled as stochastic differential equations [6]. Since fractional stochastic differential equations describe a physical dynamical system more accurately, it seems necessary to discuss the qualitative properties for such systems.
Nowadays various real-life situations can be modeled by using Poisson jumps. For example, if a system jumps from a "normal state" to " a other state", the strength of systems is random. In order to make more realistic model, a jump term is included in any dynamical systems. The study of stochastic differential equations driven by Poisson jumps has considerable attentions [9,8,5,10]. Recently, Tamilalagan et al. [10] have investigated the stochastic fractional evolution inclusions driven by Poisson jumps in a Hilbert space. Very recently Rihan et al. [11] extended to study the existence of fractional SDEs with Hilfer fractional derivative and Poisson jumps. In [12] Balasubramaniam et al. studied a class of Hilfer fractional stochastic integrodifferential equations with Poisson jumps through the fixed point technique.
The fractional Brownian motion is the usual candidate to model phenomena due to its self-similarity of increments and long-range dependence. This fractional Brownian w H is the continuous centered Gaussian process with covariance function described by The parameter H characterizes all the important properties of the process, when H < 1 2 the increments are negatively correlated and the correlation decays more slowly than quadratically; when H > 1 2 , the increments are positively correlated and the correlation decays so slowly that they are not summable, a situation which is commonly known as the long memory property. Natural candidates are the Hermite processes, these non-Gaussian stochastic processes appear as limits are called Non-Central Limit theorem [15]. The fractional Brownian motion can be expressed as a Wiener integral with respect to the standard Wiener process, i.e. the integral of a deterministic kernel with respect to a standard Brownian motion, the Hermite process of order 1 is fractional Brownian motion and of order 2 is the Rosenblatt process.
Frequently, the optimal control is largely applied to biomedicine, namely, to model the cancer chemotherapy, and recently applied to epidemiological models and medicine (see [17,18] and references therein). The main goal of optimal control is to find, in an open-loop control, the optimal values of the control variables for the dynamic system which maximize or minimize a given performance index. If a fractional differential equation describes the performance index and system dynamics, then an optimal control problem is known as a fractional optimal control problem. Using the fractional variational principle and Lagrange multiplier technique, Agrawal [13] discussed the general formulation and solution scheme for Riemann-Liouville fractional optimal control problems. It is remarkable that the fixed point technique, which is used to establish the existence results for abstract fractional differential equations, could be extended to address the fractional optimal control problems. Recently, Aicha Harrat et al. [23] studied the optimal controls of impulsive fractional system with Clarke subdifferential. Very recently, Using the Leray-Schauder fixed point theorem, Balasubramaniam et al. [3] studied the solvability and optimal controls for impulsive fractional stochastic integrodifferential equations. Tamilalagan et al. [19] investigated the solvability and optimal controls for fractional stochastic differential equations driven by Poisson jumps in Hilbert space via analytic resolvent operators and Banach contraction mapping principle. Ramkumar et al. [28] investigated the existence of mild solutions and optimal control for a class of fractional neutral stochastic differential equation driven by fractional Brownian motion and Poisson jumps in Hilbert spaces via successive approximation method.
Motivated by the aforementioned research works, in this manuscript we derive the sufficient conditions for the existence of solutions of the following class of optimal control for fractional neutral stochastic integrodifferential system driven by Rosenblatt process with Poisson jumps Hilbert space X with an inner product h·, ·i and the norm k·k. Let Y be another separable Hilbert space. The functions f :  o and the Pnull sets. Z H (t) is a Rosenblatt process with parameter H ∈ ( 1 2 , 1) on a real separable Hilbert space Y. The collection of all strongly measurable, square integrable X -valued random variable is denoted by L 2 (Ω, =, X ) ≡ L 2 (Ω, X ) which is a Banach space equipped with norm It is easy to verify that L 2 (Ω, X ) is a Banach space equipped with the above norm.
Let L(Y, X ) denotes the space of bounded linear operators from Y into X , whenever X = Y, we simply denote L(Y). Q ∈ Y represents a nonnegative self adjoint operator. We introduce the subspace and that the space L 0 2 equipped with inner product < φ, ψ > L 0

Preliminaries
In this section, we recollect basic concepts, definitions and Lemmas which will be used in the sequel to obtain the main results.
Definition 2.1. [15] The basic concepts of the Rosenblatt process as far as Wiener integral, let Z H (t) be a one-dimensional Rosenblatt process with Hurst parameter H ∈ ( 1 2 , 1). Hence the Rosenblatt process with parameter H > 1 2 representation as For basic preliminaries and fundamental results on Rosenblatt process one can refer to [15,3].
A two parameter function of the Mittag-Leffler type is defined by the series expansion where c is a contour that starts and ends with −∞ and encircles the disc |µ| ≤ |z| 1/2 counter clockwise. The most interesting properties of the Mittag-Leffler functions are associated with their Laplace integral The Caputo derivative of order α with the lower limit 0 for a function f can be written as The Laplace transform of the Caputo derivative of order α > 0 is given as

Definition 2.3. A closed and linear operator
A is said to be sectorial if there are constants ω ∈ R, θ ∈ [π/2, π], M > 0 such that the following two constants are satisfied We call A as the generator of a solution operator if there exists ω ≥ 0 and strongly continuous functions S α : S α is called the solution operator generated by A.
satisfy the uniform Holder condition with exponent β ∈ (0, 1] and A is a sectorial operator, then a continuously differentiable function x(t) is a mild solution of (1) if x satisfies the following fractional integral equation Fractional neutral stochastic integrodifferential equations with ... 555 where S α (t) is the solution operator, generated by A is given by ZB r e λt λ α−1 λ α − A dλ,B r denotes the Brownwich path [27].
is a Banach space with the norm defined by (4).

Existence of Mild Solutions
In this section, we shall derive the existence and uniqueness of mild solution for system (1). we will work under the following hypotheses: (A2) The mappings g(.), h(.) satisfy the following conditions, for all is a concave nondecreasing function from R + to R + such that κ(0) = 0, κ(ϑ) > 0 for ϑ > 0 and Let us introduce the sequence of successive approximation defined as follows x n (t) = φ(t), t ∈ [−r, 0], n = 1, 2, ...
Step 1: We first introduce two sequences of functions φ n,m (t) m,n∈N + and φ n (t) n∈N + by Then φ n (t) n∈N + is monotonically decreasing when n → ∞ and 0 ≤ φ m,n (t) ≤ φ n (t) for all m, n ≥ 1, t ∈ [0, b 1 ]. In fact, it is obvious that φ 1,m (t) ≤ φ 1 (t) and Fractional neutral stochastic integrodifferential equations with ... 561 which implies that φ 2 (t) ≤ φ 2 (t) ≤ φ 1 (t). Now assume the results holds for n, then This shows that φ n (t) is a nonnegative and decreasing continuous function on [0, b 1 ] by induction on n, so we can define a function φ n (t) by φ k (t) ↓ φ(t), and it is easy to verify that φ(0) = 0 and φ(t) is a continuous function The Borel-Cantelli lemma shows that as n → ∞, x n (t) → x(t) holds uniformly for 0 ≤ t ≤ b. So, taking limits on both sides of (5), for all −r ≤ t ≤ b, we obtain that x(t) is a solution of (1).
Step 2: Uniqueness Let x(t), y(t) be two solutions of (1). Then the uniqueness is obvious on the interval [−r, 0], and for 0 ≤ t ≤ b, it is easy to show that by using Lemma 3.2, we have The Bihari inequality yields that Therefore, x(t) = y(t) for all 0 ≤ t ≤ b.
o . Now, Consider the fractional stochastic control problem (A5) The operator B ∈ L 2 (J, L(Y, X )), kBk L 2 stands for the norm of operator B in the Banach space L 2 (J, L(Y, X )). It is obvious that Bu ∈ L 2 (J, X ) for all u ∈ U ad .
Theorem 4.1. If the hypotheses (A1) − (A5) are satisfied, for every u ∈ U ad , then there exists a unique mild solution of system (7) of the form Proof. The proof of this theorem is similar to that of Theorem 3.3, and one can easily prove that solution of system (7) by using the method of successive approximation, and hence, it is omitted. 2 To prove the existence of optimal control pair of system (7), let us define the performance index Fractional neutral stochastic integrodifferential equations with ... 563 Our aim is to find u 0 ∈ U ad such that J (u 0 ) ≤ J (u) for all u 0 ∈ U ad , where x(t) denotes the mild solution of (7), we need the following hypotheses: x, x t , ·) is concave on Y for each x t ∈ C, x ∈ X and almost all t ∈ J. (iv) There exist constants d, e ≥ 0, j > 0, ρ ≥ 0 and ρ ∈ L 1 (J, R) such that Similarly corresponding to u 0 , there exists a mild solution x 0 of (7), that is, Hence, for t ∈ J, by the hypothesis (A1) − (A5), and the Holder inequality, after an elementary calculation, we have . By lemma 4.2 in [24], B is strongly continuous and Lebesgue's dominated convergence theorem, we have For each t ∈ J, x m (.), x 0 (.) ∈ X , we have So, let us infer that x m → x 0 as m → ∞. Finally using Balder's theorem [25] and hypothesis (A6), we obtain Fractional neutral stochastic integrodifferential equations with ... 565 Hence, the result is followed that J attains its minimum at u 0 ∈ U ad .
The subordination principle of solution operator, implies that A is the infinitesimal generator of a solution operator (S α (t) t≥0 ). Because S α (t) is strongly continuous on [0, ∞) by a uniformly bounded theorem, there exist a constant M > 0 such that kS α (t)k 2 ≤ M , for t ∈ J. Define the nonlinear functions f : J × C → X , g : J × C → L 0 2 and h : J × C × Z → X by and assuming that R Z η 2λ (dη) < ∞. Furthermore, the nonlinear functions f, g and h satisfy the hypotheses (A1)−(A6). Let functions u : τ y (D) → R, such that u ∈ L 2 (τ y (D)) as the controls. This claim is that t → u(., t) going from [0, b] into Y is measurable. Set U(t) = u ∈ Y; kuk 2 Y ≤ µ, where µ ∈ L 2 (J , R + ). We restrict the admissible controls U ad to be all the u ∈ L 2 (τ y (D)) such that ku(., t)k 2 ≤ µ(t) a.c. Define B(t)u(t)x = R D k 0 (x, γ)u(γ, t)dγ and consider the following cost function −r ky(t + s, x)k 2 X dsdxdt with respect to system (10). Thus, problem (10) can be written as the form of (7). Hence, all the hypotheses stated in theorem 4.2 are satisfied. Hence, there exists an admissible control u 0 ∈ U ad such that J (u 0 ) ≤ J (u), for all u ∈ U ad . 2