Discrete Dynamics in Nature and Society
Volume 2017 (2017), Article ID 5394528, 11 pages
https://doi.org/10.1155/2017/5394528

Research Article

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

Fathalla A. Rihan,¹ Chinnathambi Rajivganthi,¹ and Palanisamy Muthukumar²

¹Department of Mathematical Sciences, College of Science, UAE University, Al-Ain 15551, UAE
²Department of Mathematics, Gandhigram Rural Institute-Deemed University, Gandhigram, Tamil Nadu 624 302, India

Correspondence should be addressed to Fathalla A. Rihan; frihan@uaeu.ac.ae

Received 18 January 2017; Revised 20 April 2017; Accepted 4 May 2017; Published 15 June 2017

Academic Editor: Thabet Abdeljawad

Copyright © 2017 Fathalla A. Rihan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. 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 [1–3]. 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 and 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

Here, is the Hilfer fractional derivative: and . is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators in Hilbert space The state variable is considered in with a norm and an inner product Let be another separable Hilbert space and is a given -valued Wiener process or Brownian motion with a finite trace nuclear covariance operator . Let be a Poisson point process in a measurable space and induced compensating martingale measure described on a complete probability space . and are appropriate functions and defines the space of all Hilbert Schmidt operators from into

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.

2. Preliminaries

Let be a complete probability space furnished with complete family of right continuous increasing sub--algebras satisfying . We assume that is the -algebra generated by and . Let and define . If , then is called a -Hilbert Schmidt operator. Let denote the space of all -Hilbert Schmidt operators . The collection of all strongly measurable, square integrable -valued random variables, denoted by , is a Banach space equipped with norm , where the expectation is defined by . Let be the Banach space of all continuous maps from into satisfying the condition Suppose that is the Poisson point process, taking its value in a measurable space with a -finite intensity measure . The compensating martingale measure and Poisson counting measure are defined by and . Let us assume that the filtration , produced by Poisson point process and is augmented, where is the class of -null sets.

Define = ; and let

Obviously, is a Banach space.

Definition 1. The fractional integral of order with the lower limit for a function is defined as

provided that the right-hand side is pointwise defined on , where is the Gamma function.

Definition 2 (see [5]). The Hilfer fractional derivative of order and 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].

Remark 3. When , the Hilfer fractional derivative coincides with classical Riemann-Liouville fractional derivative:

When , the Hilfer fractional derivative coincides with classical Caputo fractional derivative:

For , let us define the operators and by

where , is a function of Wright-type defined on and verifies , and

Lemma 4 (see [20]). The operators and have the following properties:

(i)For any fixed and are bounded and linear operators, and and .(ii) is compact, if is compact.

Definition 5 (see [19, 20]). An -valued stochastic process is a mild solution of system (1) if the process satisfies the following integral equation:

Remark 6. (i)
(ii) When , the fractional stochastic equation (1) simplifies to the classical Caputo fractional equation which has been discussed by Chen and Li [25]. In this case, , where is defined in [25].

We impose the following assumptions to show the main results:

The maps , , and satisfy, for all and ,

 where is a concave nondecreasing function from to such that , for and For all , there exists a constant such that

The reader may refer to Remark and Lemmas and in [30], which are useful to prove the main results.

Let the solution of (1) be defined as follows:

We refer to [25, 38, 39] for further discussion of stochastic concepts.

3. 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 are satisfied, then system (1) has a unique mild solution in the space , provided that , with and

Proof. For better readability, we break the proof into a sequence of steps.
Step 1. For all , the sequence , is bounded.
It is obvious that Let be a fixed initial approximation to (12). Let us use the assumptions and , Holder inequality, Doob Martingale inequality and Burkholder-Davis-Gundy inequality for pure jump stochastic integral in ([30]). We have

where and is constant. Here, is concave and , and one can find a pair of positive constants and such that , for . Then

where

For any ,

where and . Thus , for , , which shows that the sequence , is bounded in .
Step 2. Sequence , is a Cauchy sequence.
From (12), for all and ,

Let Thus, we have in the above inequality that

Choose such that

Moreover,

We take the supreme over and use :

Now, for in (18), we have

And, for in (18), we have

By applying mathematical induction in (18) and with the above work, we have

So, for any ,

Step 3. The existence and uniqueness of solution for system (1) are tackled as follows.
The Borel-Cantelli Lemma says that , as uniformly for Thus, for all , taking limits on both sides of (12), one can obtain that is a solution to (1). Next, to show the uniqueness, let be two solutions on For ,

Thus, from Bihari inequality, it yields that

Therefore, , for all

4. 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 The multivalued function is measurable and , where is a bounded set of The admissible control set Then and is bounded, closed, and convex [35]. The fractional stochastic control problem with Hilfer fractional derivative is of the form

By using Definition 5 for every , there exists , and the solution of the control problem (28) is defined as

The operator ; denotes the norm of operator in Banach space . Obliviously, for every .

Lemma 8. Let hold. If system (28) is mildly solvable on with respect to and , then there exists a constant such that for all

Proof. If is a mild solution of (28) with respect to , then satisfies equation (29). Using hypotheses , as well as Burkholder-Davis-Gundy inequality ([30]), we obtain

where + + ) and . Given that is concave and , one can find a pair of positive constants and such that , for Then

By using Gronwall’s inequality,

Theorem 9 (see [35]). Under hypotheses and for each , system (28) is mildly solvable on and the solution is unique.

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 such that

here defines the mild solution of (28) corresponding to . Assume that

the cost functional is such that

(i) is measurable,(ii) is lower semicontinuous on for almost all ,(iii) is convex on for all and almost all ,(iv)there exist constants , , , and such that

Theorem 10. If is strongly continuous operator, hypotheses and Theorem 9 hold. Then the stochastic control problem (28) permits at least one optimal control pair.

Proof. The main aim is to minimize the value of If , ( = such that is a mild solution of (28) with ); then there is nothing to prove. Assume that Using , we have By definition of infimum, there exists a minimizing sequence feasible pair , such that as Since , is bounded. Thus, there exists and a subsequence extracted from (still called ) such that weakly in Moreover, from the convexity and closeness of and Mazur’s Theorem, we know that Suppose that and are the mild solutions of (28) corresponding to and , respectively. and satisfy the following equations, respectively:

From the boundedness of and , Lemma 8, one can verify that there exists a positive number such that Then, for .

Using the continuous operator and Lebesgue’s dominated convergence theorem, we have

Consequently, as Thus, by and Balder’s theorem [40], we obtain

This shows that attains its minimum at

5. Example

In this section, we provide an example to verify the theoretical results. Consider the control problem

Here, is the Hilfer fractional derivative, Let denote a standard one-dimensional Wiener process in defined on . The operator is defined by with the domain . Then , , where , , is the orthogonal set of eigenvectors of . It is well known that generates a compact semigroup in and is given by , for Moreover, for any , we have

The Poisson point process induced the Poisson counting measure and the compensating martingale measure defined as

The nonlinear functions , and are defined by

and assuming that , . Clearly, the functions , and satisfy the assumptions . If , 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). The controls are the functions , such that It means that going from into is measurable. Set , where . We restrict the admissible controls to be all such that almost everywhere.

Let us define and make the following assumptions:

(i) is continuous.

(ii) and is defined by

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

attains its minimum.

6. 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.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research was generously supported by the UAE University.

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