Fractional Stochastic Differential Equations with Hilfer Fractional Derivative : Poisson Jumps and Optimal Control

In this work, we consider a class of fractional stochastic differential system with Hilfer fractional derivative and Poisson jumps in Hilbert space. We study the existence and uniqueness of mild solutions of such a class of fractional stochastic system, using successive approximation theory, stochastic analysis techniques, and fractional calculus. Further, we study the existence of optimal control pairs for the system, using general mild conditions of cost functional. Finally, we provide an example to illustrate the obtained results.


Introduction
The subject of fractional calculus has gained importance and attractiveness due to its applications in widespread fields of engineering and science.Fractional calculus is successful in describing systems which have long-time memory and long-range interaction [1][2][3].Fractional-Order Differential Equations (FODEs) models have been successfully applied in biology systems [3,4], physics [5,6], chemistry and biochemistry [7], hydrology [8], engineering [9,10], medicine [11], finance [12], and control problems [13,14].In most cases, the models of FODEs seem to be more regular with the real events compared with integer-order models, because fractional integrals and derivatives allow the explanation of the hereditary and memory properties inherent in various processes and materials [15,16].Many authors described the fractional-order models with the most common definitions of fractional derivatives defined by Caputo and Riemann-Liouville sense [17].
Hilfer [5] proposed a general operator for fractional derivative, called "Hilfer fractional derivative," which combines Caputo and Riemann-Liouville fractional derivatives.Hilfer fractional derivative is performed, for example, in the theoretical simulation of dielectric relaxation in glass forming materials.Sandev et al. [18] derived the existence results of fractional diffusion equation with Hilfer fractional derivative which attained in terms of Mittag Leffler functions.Mahmudov and McKibben [19] studied the controllability of fractional dynamical equations with generalized Riemann-Liouville fractional derivative by using Schauder fixed point theorem and fractional calculus.Recently, Gu and Trujillo [20] reported the existence results of fractional differential equations with Hilfer derivative based on noncompact measure method.The set of two parameters in "Hilfer fractional derivative"  ],  + of order 0 ≤ ] ≤ 1 and 0 <  < 1 permits one to connect between the Caputo and Riemann-Liouville derivatives [17,21,22].This set of parameters gives an extra degree of freedom on the initial conditions and produces more types of stationary states.Models with Hilfer fractional derivatives are discussed in [23,24].
The deterministic models often fluctuate due to noise.Naturally, the extension of such models is necessary to consider stochastic models, where the related parameters are considered as appropriate Brownian motion and stochastic processes.The modeling of most problems in real situations is described by stochastic differential equations rather than deterministic equations.Thus, it is of great importance to design stochastic effects in the study of fractional-order dynamical systems.Chen and Li [25] reported the existence results of fractional stochastic integrodifferential equations with nonlocal initial conditions in Hilbert space.Wang [26] investigated the existence results of fractional stochastic differential equations by using Picard type approximation.Lu and Liu [27] studied, recently, the controllability of fractional stochastic hemivariational inequalities based on multivalued maps and fixed point theorem.The above-mentioned research papers discussed the detail of stochastic differential equations (SDEs) with Brownian motion, Although Brownian motion cannot be used to define the stochastic disturbances in some real systems such as the fluctuations in the financial markets and price dynamics of financial instruments with jumps (see [28]).The authors in [29] studied the existence results of jumps in stock markets and the foreign exchange markets which are based on SDEs with Poisson jumps.Ren et al. [30] reported the existence and stability results of time-dependent stochastic delayed differential equations with Poisson jumps.Recently, Rajivganthi and Muthukumar [31] studied the properties of almost automorphic solutions of fractional stochastic evolution equations with Poisson jumps with the help of solution operator.
To the best of our knowledge, the existence and uniqueness of mild solutions for fractional stochastic differential equations with Hilfer fractional derivative are an untreated topic in the present literature.Herein, we convert the deterministic fractional differential equations into a stochastic fractional differential equation with Hilfer fractional derivative.We then study the existence and uniqueness of mild solutions by using successive approximation.We study the existence and uniqueness of mild solutions by using successive approximation theory.This theory possesses some advantages of linearization for the nonlinear functional with respect to the state variables.We then study the existence of optimal control pairs for the system, using general mild conditions of cost functional.
Consider the fractional stochastic differential equations with Hilfer fractional derivative and Poisson jumps of the form  ], (1)
Frequently, the optimal control problems stand up in system engineering.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.The determination of optimal control is a difficult task and is open-ended due to the nonlinear nature of dynamic systems.If the FODEs are described in conjunction with a set of initial conditions and performance index, they become Fractional Optimal Control Problems (FOCPs).The FOCP refers to optimization of the performance index subject to dynamical constraints on the control and state which have fractional-order models.There has been some work done in the area of deterministic FOCPs in finite dimensional spaces [32,33] and infinite dimensional cases [34,35].Ren and Wu [36] discussed the optimal control problem associated with multivalued SDEs with Levy jumps by using Yosida approximation theory.Ahmed [37] studied the existence and optimal control of stochastic initial boundary value problems subject to boundary noise.Rajivganthi et al. [38] investigated the optimal control results of fractional stochastic neutral differential equations in Hilbert space.Motivated by the work done by the authors [20,35,38], in this paper, we study additionally the sufficient conditions that guarantee the optimal control results for the fractional stochastic system (1).
This paper is prepared as follows.In Section 2, we provide some remarks, definitions, and lemmas which are useful to prove the main results.Suitable sufficient conditions for existence and uniqueness of (1) are studied in Section 3. Optimal control results are discussed in Section 4.An example is given in Section 5 to verify the theoretical results.We then conclude the paper in the last Section.
Definition 1.The fractional integral of order  > 0 with the lower limit  for a function  : [, ∞) → R is defined as provided that the right-hand side is pointwise defined on [0, ∞), where Γ is the Gamma function.
Definition 2 (see [5]).The Hilfer fractional derivative of order 0 ≤ ] ≤ 1 and 0 <  < 1 with lower limit  is defined as for functions such that the expression on the right-hand side exists.
For more details about the Caputo and Riemann-Liouville fractional derivatives, the reader may refer to [22].
We impose the following assumptions to show the main results: ( 1 ) The maps  :  ×  → ,  :  ×  →   (, ), and ℎ :  ×  ×  →  satisfy, for all  ∈  and  1 ,  2 ∈ , where K(⋅) is a concave nondecreasing function from ( 2 ) For all  ∈ , there exists a constant  0 > 0 such that The reader may refer to Remark 2.3 and Lemmas 2.4 and 2.5 in [30], which are useful to prove the main results.

Existence and Uniqueness of Mild Solutions
In order to prove the existence of mild solution for system (1), let us consider the sequence of successive approximations defined as follows: Theorem 7. If the assumptions ( 1 )-( 2 ) are satisfied, then system (1) has a unique mild solution in the space  ], (,  2 (Ω; )), provided that Proof.For better readability, we break the proof into a sequence of steps.

Optimal Control Results
Let  be reflexive Banach space in which controls  take values.Let us denote a class of nonempty convex and closed subsets of  by 2  \ {Ø}.The multivalued function  Obliviously,  ∈  2 (, ) for every  ∈  ad .
Minimize a performance index of the following form: among all the admissible state control pairs of system (28); that is, find an admissible state control pair ( 0 ,  0 ) ∈ (,  2 (Ω; )) ×   such that J ( 0 ,  0 ) ≤ J (, ) ∀ ∈   ; here   () defines the mild solution of (28) From the boundedness of   and û, Lemma 8, one can verify that there exists a positive number  such that ‖  ‖, ‖ x‖ ≤ .
If  = 0, then problem (41) can be written as the form of (1).All the conditions stated in Theorem 7 are satisfied for system (41) and can be applied to ensure the existence and uniqueness of the mild solution of (41  (45) Then, system (41) can be written as in the form of (28).All the conditions stated in Theorem 10 are verified.Therefore, there exists an admissible control pair (, ) such that the associated cost functional J (, ) = ∫  0 L (,   () ,  ())  (46) attains its minimum.

Concluding Remarks
In this paper, we studied the existence of solutions and optimal control results of fractional stochastic differential equations with Hilfer fractional derivative and Poisson jumps.The existence and uniqueness of mild solutions for the system have been obtained by using the successive approximation theory and stochastic analysis techniques.New sufficient conditions for optimal control results of fractional stochastic control system have been deduced.Throughout an example, the effectiveness of the obtained results is proven, under suitable conditions, for fractional stochastic partial differential equations with Poisson jumps.The optimal control analysis for fractional stochastic differential inclusions with distributed delays, time varying delays, and impulsive effects will be our future work.