Salah H. Abid, Sameer Q. Hasan, Uday J. Quaez
Mathematics department, Education College, Al-Mustansiriya University, Baghdad, Iraq
Correspondence to: Salah H. Abid, Mathematics department, Education College, Al-Mustansiriya University, Baghdad, Iraq.
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Copyright © 2015 Scientific & Academic Publishing. All Rights Reserved.
Abstract
In this paper, we will study the Approximate controllability of fractional stochastic integro-differential equations which is derived by mixed type of fractional Brownian motion with Hurst parameter and wiener process in real separable Hilbert space. An example was stated as a application of our result.
Keywords:
Approximate controllability, Mixed type of fractional Brownian motion, Fixed point contraction principle, Stochasticintegro- Differential equations, Mild solution, Control function
Cite this paper: Salah H. Abid, Sameer Q. Hasan, Uday J. Quaez, Approximate Controllability of Fractional Stochastic Integro-Differential Equations Driven by Mixed Fractional Brownian Motion, American Journal of Mathematics and Statistics, Vol. 5 No. 2, 2015, pp. 72-81. doi: 10.5923/j.ajms.20150502.04.
1. Introduction
The purpose of this paper is to prove the existence and approximate controllability of mild solution for the class of fractional stochastic integro-differential equations driven by mixed type of fractional Brownian motion with Hurst and wiener process. The following form is the system under our consideration, | (1) |
where the Riemann-Liouville fractional derivative of order α. A: Dom(A) ⊂ X →X is the infinitesimal generator of strongly continuous compact semi group of bounded linear operators in X. x(.) takes the value in the real separable Hilbert space X such that for each, the banach space of all continuous functions from [0,T] in to satisfying the condition and is a banach space of all F-measurable square integrable random variables with values in Hilbert space X equipped with the sup norm F0 –measurable X- valued random variable independent of W and is the space of the Ft – adapted , U-valued measurable process u(t) on [0,T] such that with norm where U is a real separable Hilbert space. B is the linear bounded operator from U into X such that there exists constant is a Q-fractional Brownian motion with Hurst index defined in a complete probability space with values in a Hilbert space Y. is a Q-l wiener process defined on with values in a Hilbert space K.and are continuous functions and uniformly bounded. is a deterministic function the processes W and are independent.In the past few decades, the theory of fractional partial differential equations in both types deterministic and stochastic, have received a great deal of attention and play an important role in many applied scientific fields.The deterministic models often affected due to fractal noise, which is random or at least appears to be so. Therefore, the study of stochastic systems are more applicable in dynamical system theory. Random phenomena exist everywhere in the real world. Systems are often.Subjected to random perturbations. The existence of solution for some classes of Stochastic equations driven by fractional Brownian motion have been investigated by many authors, see, for example [4], [9], [19].The controllability of stochastic differential equations in infinite dimensional spaces have been investigated by many authors [6], [7], [8], [11], [12]. In recent years, sakthivel [15] derived a set of sufficient conditions for approximate controllability of nonlinear impulsive stochastic differential equations in a real separable Hilbert space by using the stochastic analysis theory and a fixed point. Zang and Li [20] studied the approximate controllability of fractional impulsive neutral stochastic differential equations with non-local conditions and infinite delay in Hilbert space. Guendouzi [18] studies the approximate controllability result of a class of dynamic control systems described by nonlinear fractional stochastic functional differential equations in Hilbert.Space driven by a fractional Brownian motion with Hurst parameter H > 1/2 by using the theory of fractional calculus and a fixed point theorm. Hamdy [5] studied the approximate controllability for impulsive neutral stochastic functional differential equations with finite delay and fractional Brownian motion in a Hilbert space by using semigroup theory, stochastic analysis, and Banach’s fixed point theorem.In this paper we will study the approximate controllability of nonlinear stochastic system. More precisely, we shall formulate and prove sufficient conditions for the Approximate controllability of fractional stochastic integro-differential equations driven by mixed type of fractional Brownian motion with Hurst and wiener process in Hilbert space. The rest of this paper is organized as follows, in section 2, we will introduced some concepts, definitions and some lemmas of fractional stochastic calculus which are useful for us here. In section 3, we will prove our main result. Finally in section 4, as an application, an example will be stated in details.
2. Preliminaries
In this section, we introduce some notations and preliminary results, needed to establish our results. Throughout this paper, let X, Y, K be real separable Hilbert spaces and be a complete probability space with natural filtration generated by therandom variables and the P-null set). We denote by L (K; X) the space of all bounded linear operators from K to X and L(Y; X) denote the space of all bounded linear operators from Y to X. For convenience we will use the same notation to denote the norms in K, Y, X,L (K;X), L(Y;X) and use to the inner product of K , Y , X.Definition (2.1) [1]:A standard fractional Brownian motion with Hurst index H ∈ (0,1) is a Gaussian process on (Ω, F, P) having the properties i. ii. iii. If Then the increments of B^{H} are non-correlated, and consequently independent. So B^{H} is a standard Wiener Process which we denote further by B.-If H then the increments are positively correlated-If H then the increments are negative correlatedMoreover, B^{H}^{ }^{ }has the integral representationWhere B is a standard Wiener process and the kernel defined asand in the case we shall use Ito Isometry | (2) |
for every the class of functions such that and f is measurable, adapted and Suppose We denote by the set of step functions on [0, T]The integral of Φ∈ᶓ with respect to a standard fractional Brownian motion will be defined by, Where NOW, Let be the Hilbert space defined as the closure of ᶓ with respect to the scalar productThe mapping can be extended to an isometry between H and Let │H│ be the Banach space of measurable functions ' on [0, T], such thatLemma (2-1) [9] and for any we haveNow Suppose that there exists a complete orthonormal system in Y. Let be the operator defined by where (n=1,2,….) are non-negative real numbers .With finite trace.Tr Analogically to Wiener processes in a Hilbert space, we defined A fractional Brwnian Motion on Y by using covariance operator Q as Where are standard fractional Brownian motions mutually independent on In order to defined stochastic integral with respect to the Q-fractional Brownian motion.We introduce the space of all Q-Hilbert- Schmidt operators that is with the inner product _{ }is a separable Hilbert space. L2(K;X) the space of all Hilbert-Schmidt operators acting between K and X equipped with the Hilbert-Schmidt norm Lemma (2-2)Let be a deterministic function with values in The stochastic integral of Φ with respect to is defined by | (3) |
Lemma (2-3) [9]If satisfies then the above sum in (3) is well defined as an X-valued random variable and we have | (4) |
Definition (2-2) [15]:- The fractional integral of order with the lower limit 0 for a function f is defined as:Where is a gamma function.Definition (2-3) [15]:- The Riemann - Liouvill derivative of order with lower zero for a function can be written as:Definition (2-4) [15]:- The Laplace transform of the Riemann-Liouville fractional integral gives Where Definition (2-5) An X-valued process x(t) is called a mild solution of the stochastic integro-differential equation with mixed type of Brownian motion which is Wand are independent in (1) If | (5) |
where is a Mainardi's functionWhere Lemma (2-4):- is a strongly continuous compact semigroup of bounded linear operators in X, then The operator have the following properties:(i) For any fixed is a linear and bounded operator, i.e, | (6) |
(ii) is a strongly continuous, which mean that for every and we have (iii) For every is compact operator.Proof: The proof of this lemma similar to the proof of the Lemma 3.2 (see [21]).In order to study the approximate controllability for fractional stochastic control system (1) we introduce the following linear fractional differential system corresponding to system (1) | (7) |
Definition (2-6)The set (where x(T;x_{0},u) the state value of the system (1) at time terminal time T corresponding to the control u and the initial value x0) is called the reachable set of system(1) at terminal time T. The closure of R (T, x_{0}) in the space is denoted by R (T, x_{0}).Definition (2-7) The system (1) is said to be approximately controllability on [0,T] if the reachable set R (T,x_{0}) is dense in the space this mean that Lemma (2-5) [12] The linear fractional deterministic system (2.9) is approximately controllable on [0, T] iff the operator for all and Moreover Lemma (2-6) For any there exists and Such that | (8) |
We Define the operator as | (9) |
It is clear that is bounded if The adjoint operator is defined by Now to defined the controllability operator associated with (1) as | (10) |
and the controllability operator associated with(7) as | (11) |
clearly that the operators are linear bounded.For any and we defined the operator The relationship between controllability operator and is (see [7], [12]) | (12) |
| (13) |
Definition (2-8) for any and any the stochastic control function of the system (2.1) in the following form: | (14) |
3. Main Result of the Approximate Controllability
In this section, we will formulate the sufficient conditions and prove the result for the approximate controllability of nonlinear fractional stochastic system (1). For this purpose, firstly, we will prove the existence and uniqueness of solution by using the contraction mapping principle. Secondly, we shall prove in theorem 3.2, that the system in (1) is approximate controllability under certain assumptions. Now, assume that, (H1): for The operator is compact and satisfies (H2) The linear fractional differential system (7) is approximately controllable on [0,T].(H3) The function satisfies the following properties(i) (ii) (iii) G is Lipschitz condition for all there exists such that (H4) The function satisfies (H5) There exists a positive constant such that (H6)The function is a continuous and satisfies the usual growth condition and Lipchitz condition: there exist constantLemma 3-1There exists positive real constants and such that, for all | (15) |
| (16) |
Proofi. Let be a fixed from (14) we have By using the inequality, lemma 2.4, Ito isomery theorem and from lemma 2.3 we obtainSo, from the assumptions of data, we obtain Where Since the proof of the second inequality can be verified in a similar manner. The proof is completed.Theorem (2-1)Assume that the conditions (H1) – (H6) are satisfied, then the system (1) has a mild solution on [0, T].Proof: For any define the operator by | (17) |
It will be shown that the system (1) is approximately controllable if for all There exists a fixed point of operator To prove this result, we use the contraction mapping principle.The proof of the theorem is long and technical, therefore it is convenient to divide it in to three steps.Step 1To prove for any is continuous on [0,T] in Let such that t_{1}< t_{2}. Then for any fixed we have Now By using the strong continuity of a semigroup with lemma 2.4 we get if and by using Lebesgueś dominated convergence theorem , we conclude that the right –hand side of the above inequality tend to zero as Thus we conclude is continuous from the right in [0,T).A similar argument shows that it is also continuous from the left in (0,T].Thus is continuous on [0,T] in Step 2For each To show that the operator maps into itself. Let by using inequality, lemma 2.4, Ito isomery theorem, lemma 2.3 and assumptions H1-H6 we obtainFrom this inequality We get this mean that.Step 3:In both steps 1 and 2 we showed that the operator is a continuous on [0,T] and so maps from in to In this step we will prove the theorem through the Banach fixed point theorem that for each fixed the operator has a unique fixed point in Now, we will show that is a contraction mapping in Let then for any fixed t ∈ [0,T] we haveUsing inequality, lemma 2.4, Ito isomery theorem, lemma 2.3, assumptions H1-H6 and the fact that we obtainTaking the supremum over [0, T] for both sides of the above inequality, we get Then, WhereSo, there exists such that and is a contraction mapping on and therefore has a unique fixed point, which is a mild solution of equation 2.1 on [0, T_{1}]. This procedure can be repeated in order to extend the solution to the entire interval [0,T] in finitely many steps. This completes the proof.Theorem (2-3)Assume the conditions (H1)-(H6) are satisfied and moreover assume that F and G are uniformly bounded. Then the system (2.1) is an approximately controllable on [0,T].Proof for every let be a fixed point of the operator in From (17) we have | (18) |
Using the control function in (14), the stochastic Fubini theorem and the definition of controllability operator in…(11), we obtain Since the functions F and G are uniformly bounded, then, there exists constants Ď1> 0 and Ď2 > 0 such that In X. and in L2(K ;X)Where So, the sequences andare bounded in X and L2 (K;X) respectively. Then, there are subsequences denoted by and that converges weakly toF(s) in X and G(s) in L2 (K;X) respectively.Now since is compact by lemma 2.4 then in X andOn the other hand by the assumption (H2), for all the operator Strongly as and moreover NowUsing Ito isometry and lemma 2.3 we have By the Lebesgue dominated convergence theorem, the compactness of and strongly as for all 0 and moreover we obtainThis gives the approximate controllability.
4. Illustrative Example
In this section we will take the following example as an application to theorem 2.2Consider the following control system governed by the stochastic fractional partial integro-differential equation, | (19) |
To study this system, let {all square integralable functions on with values in real numbers R} Then F is continuous and uniformly bounded function DefineHere g is a continuous and uniformly bounded function.Let be an operator defend by A with domain are absolutely continuous, From this, A is well defined that it is infinitesimal generated of compact semi group in X and it is given by S(t) Where is a complete orthonormal basis in X, From this, we have With the choice of A, F, G and h, (19) can be rewritten as the form of system (1). Thus, under the appropriate assumptions on the functions F, G, h as those in H1 –H6, system (19) is approximately controllable .
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