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Approximate Controllability of Fractional Sobolev Type Stochastic Differential Equations Driven by Mixed Fractional Brownian Motion

Salah H. Abid*, Sameer Q. Hasan, Uday J. Quaez

Mathematics department, Education College, Al-Mustansiriya University, Baghdad, Iraq

Abstract: In this paper, the approximate controllability of nonlinear Fractional Sobolev type with order Caputo stochastic differential equations driven by mixed fractional Brownian motion in a real separable Hilbert spaces has been studied by using contraction mapping principle, fixed point theorem, stochastic analysis theory, fractional calculus and some sufficient conditions.

Keywords: approximate controllability, mixed fractional brownian motion, fixed point contraction principle, stochastic differential equations, mild solution, control function

Received June 25, 2015; Revised August 03, 2015; Accepted August 14, 2015

Citation: Salah H. Abid, Sameer Q. Hasan, Uday J. Quaez, Approximate Controllability of Fractional Sobolev Type Stochastic Differential Equations Driven by Mixed Fractional Brownian Motion. Journal of Mathematical Sciences and Applications, 3 (1). 3-11. January 2015. doi:10.12691/jmsa-3-1-2

1. Introduction

The purpose of this paper is to prove the existence and approximate controllability of mild solution for the class of fractional Sobolev type stochastic differential equations driven by mixed fractional Brownian motion with Hurst and wiener process. The following form is the system under our consideration,

[picture] | (1)

where, equipped with the sup norm such that X is a real separable Hilbert space. the Caputo fractional derivative of order and the Riemann-Liouville fractional derivative of order . The operators and L are defined on domains contained in X, and ranges contained in a real separable Hilbert space Z, such that is a bijective linear operator, is a compact and L is a closed linear operator. The control function is a Hilbert space and the operator B from U into Z is a bounded linear operator. The functions , , , and are continuous functions such that K and Y are real separable Hilbert spaces.

are the standard cylindrical Brownian motion (cylindrical wiener process) defined on complete probability space equipped with normal filtration is the sigma algebra generated by

Let Q be a positive, self –adjoint and trace class operator on K and let L₂ (K,X) be the space of all Q -Hilbert-Schmidt operators acting between K and X equipped with the Hilbert-Schmidt norm

and are the Q-fractional Brownian motion with Hurst index defined in a complete probability space with values in a real separable Hilbert space Y, such that Q is a positive, self –adjoint and trace class operator on Y and let be the space of all Q -Hilbert-Schmidt operators acting between Y and X equipped with the Hilbert-Schmidt norm , and are independents which defined on a complete probability space

Approximate controllability of stochastic differential equations driven by fractional Brownian motion has been interested by many authors; Sakthivel [19] referred to future study for the approximate controllability of impulsive stochastic systems with fractional Brownian motion. Guendouzi and Idrissi, [7] established and discussed the approximate controllability result of a class of dynamic control systems described by nonlinear fractional stochastic functional differential equations in Hilbert space driven by fractional Brownian motion with Hurst parameter . Ahmed [2] investigate the approximate controllability problem for the class of impulsive neutral stochastic functional differential equations with finite delay and fractional Brownian motion with Hurst parameter in a Hilbert space. Abid, Hasan and Quaez [1] studied the Approximate controllability of fractional stochastic integro-differential equations which is derived by mixed type of fractional Brownian motion with Hurst parameter H >12 and wiener process in real separable Hilbert space. On the other hand, Sobolev type differential equations have been investigated by many authors, for example, Balachandran, Kiruthika and Trujillo [3] established the existence of solutions of nonlinear impulsive fractional integrodifferential equations of Sobolev type with nonlocal condition. Zhou, Wang and Feckan [20] investigated a class of Sobolev type semilinear fractional evolution systems in separable Banach space. Kerboua, Debbouche and Baleanu [8] sudied the approximate controllability of Sobolev type fractional stochastic nonlocal nonlinear differential equations in Hilbert spaces.

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 Sobolev type stochastic differential equations driven by mixed 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.

2. Preliminaries

In this section, we introduce some notations and preliminary results, which we needed to establish our results.

Definition (2.1), [5]:

Let H be a constant belonging to (0, 1). A one dimensional fractional Brownian motion of Hurst index H is a continuous and centered Gaussian process with covariance function

[picture] | (1)

•  If , then the increments of BH are non-correlated, and consequently independent. So BH is a Wiener Process which we denote further by B.

•  If then the increments are positively correlated.

•  If then the increments are negative correlated.

B H has the integral representation

[picture] | (2)

where, B is a wiener process and the kernel defined as

[picture] | (3)

[picture] | (4)

and is a beta function.

In the case , we shall use Ito Isometry theorem

Lemma (2.1), “Ito isometry theorem”, [11]:

Let V [0,T] be the class of functions such that is measurable , adapted and Then for every we have

[picture] | (5)

where B is a wiener process.

Now, we denote by the set of step functions on [0, T]. If Φ ∈ then, we can write it the form as:

The integral of a step function Φ∈ with respect to one dimensional fractional Brownian motion is defined

where

Let be the Hilbert space defined as the closure of with respect to the scalar product The mapping can be extended to an isometry between and .i.e. the mapping is isometry.

Remark (2.1):

•  If and then by use Ito isometry, we have

[picture] | (6)

•  If , we have

[picture] | (7)

[picture] | (8)

Lemma (2.2), [6]:

For any functions Φ, , we have

i)

ii)

From this Lemma above, we obtain

[picture] | (9)

Remark (2.2), [6]:

The space contains the set of functions , such that, , which includes

Now,

Let be the Banach space of measurable functions on [0, T], such that

[picture] | (10)

Lemma (2.3), [10]:

Suppose that there exists a complete orthonormal system in Y. Let be the operator defined by , where are non-negative real numbers with finite trace . The infinite dimensional fractional Brownian motion on Y can be defined by using covariance operator Q as

where are one dimensional 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.

Lemma (2.4), [10]:

Let be a deterministic function with values in The stochastic integral of with respect to is defined by

[picture] | (11)

Lemma (2.5), [10]:

If satisfies then the above sum in (11) is well defined as an X-valued random variable and we have

[picture] | (12)

Definition (2.2), [18]:

The Riemann - Liouvill derivative of order with lower limit zero for a function f can be written as:

[picture] | (13)

where,

Definition (2.3), [18]:

The Caputo derivative of order with lower zero for a function f can be written as:

[picture] | (14)

where, .

Remark (2.3), [9]:

The relationship between the two definitions Riemann – Liouvill derivative and Caputo derivative gives as:

[picture] | (15)

where, .

Definition (2.4), [18]:

The Laplace transform of the Riemann-Liouville fractional derivation of order gives as:

[picture] | (16)

where,

Definition (2.5), [18]:

The Laplace transform of the caputo derivation of order is given as:

[picture] | (17)

where,

Remarks (2.4)

i) The operator is a bijective linear operator, then is a a bijective linear

ii) is a compact linear operator, we obtain that is bounded.

iii) is a bounded and L is a closed linear operator by (closed graph theorem), we obtain the boundedness of linear operator

vi) The operator is bounded. Then, is an infinitesimal generator of semigroup Suppose that <∞. (see[4])

Definition (2.6):

An X-valued process is a mild solution of the stochastic differential equation driven by mixed fractional Brownian motion in (1) if, for each control function the integral equation

[picture] | (18)

is satisfied.

where, the operators and are given by

[picture] | (19)

[picture] | (20)

is a Mainardi's function.

Lemma (2.6):

If is a strongly continuous semigroup by linear operator then the operators and have the following properties:

i. For any fixed , the operators and are linear and bounded, i.e. for any , there exists > 1 such that

where ,

ii. The operatorsand are strongly continuous, which mean that for every and we have

iii. is a compact operator in X for each

3. Main Result of the Approximately Controllable

In this section, we formulate and prove the result on approximate controllability of nonlinear fractional Sobolev type stochastic differential equations driven by mixed fractional Brownian motion in (1). To establish our results, we introduce the following assumptions:

a) The semigroup which generated by linear operator is a strongly continuous and it is compact for any .

b) The Sobolev type linear fractional order Caputo type system of corresponding to the system (1) of the following form:

[picture] | (21)

is approximately controllable on [0,T].

c) The function satisfies : for every , and there exists such that .

d) The function satisfies : for every and there exists such that

e) The functions, and satisfy linear growth and Lipschitz conditions. This mean that, for any , there exists positive constants and such that

Definition (3.1):

The system (1) is said to be approximately controllable on [0,T] if the reachable set is dense in the space This mean that where,

Remark (3.1), [12]:

The linear fractional order system (21) of the corresponding system (1) is a natural generalization of approximate controllability of linear first order control system.

The controllability operator associated with equation (21) is defined by

[picture] | (22)

Also, for any and , the operator is defined by

[picture] | (23)

where, and are the adjoint operators for B and respectively.

Lemma (3.1), [12]:

The Sobolev type linear fractional order deterministic system in (21) is an approximately controllable on [0, T] if and only if the operator . Moreover .

Lemma (3.2), [13]:

For any there exists and

such that

[picture] | (24)

where, ,

Now, For any and any we define the control function of the system (1) in the following form:

[picture] | (25)

Lemma (3.3):

There exists positive real constants such that for all we have

i

[picture] | (26)

ii

[picture] | (27)

Proof

i. Let and be a fixed. From the equation (25), we have:

Applying Holder’s inequality and from Ito isomerty theorem, we obtain

From the assumption (e), we get

The proof of ii. similar to the proof of the Lemma (3.1) (see [1]).

Now, for any , we define the operator on the space by

[picture] | (28)

Lemma (3.4):

For any the operator is a continuous on [0, T] in the space .

Proof:

Let such that Then for any , from (28), we have

Applying Holder’s inequality on the last inequality and by using Ito isometry, Lemma (2.5), Lemma (3.3), Lemma (2.6) and Lemma (3.1), we obtain

Hence, by using the strong continuity of and in Lemma (2.6) and Lebesgue’s dominated convergence theorem, we conclude that the right-hand side of the above inequalities tends to zero as . Thus, we conclude is a 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 the

Lemma (3.5):

For each , the operator maps from into itself. i.e.

Proof:

From Lemma (3.4), for any , the operator is a continuous on [0, T] in the space . To prove that for implies .

Applying Holder’s inequality and by using Ito isometry theoem, Lemma (2.5), Lemma (3.3), Lemma (2.6) and Lemma (3.1), we obtain

Hence, the last inequality imply that . Moreover, for then Thus for each , the operator maps from into itself.

Theorem (3.1):

Let the assumptions (a)-(e) be satisfied. Then for each the system (1) has a mild solution on [0, T].

Proof:

To prove the existence of a fixed point of the operator which is defined in (28) by using the contraction mapping principle.

Let . for any , we have

Applying Holder’s inequality on the last inequality, by using Ito isometry and Lemma (2.5), we obtain

From the assumptions (a)-(e) and Lemma (3.3), we obtain

Taking supremum over for both sides, we get

where,

Then, there exists such that and is a contraction mapping on and therefore has a unique fixed point, which is a mild solution of equation (1) on This procedure can be repeated in order to extend the solution to the entire interval in finitely many steps. This completes the proof.

Theorem (3-2):

Assume that the assumptions (a) – (e) are satisfied, Further, if the functions F, G₁ and are uniformly bounded, then the system (1) is approximately controllable on [0, T].

Proof:

For every , let be a fixed point of the operator in which is a mild solution under the control function in (25) of the system (1). Then from (28), we have:

[picture] | (29)

It follows from the assumptions on F, G₁ and G₂ that there exists such that, , , , for all Then, there is a subsequences still denoted by and which converges weakly to and in and respectively.

Now, from the equation (29), we have

Using Ito isometry and Lemma (2.5) and Lemma (4-2), we obtain

On the other hand, by the assumption (b), and Lemma (3.1), for all , we have the operator strongly as and moreover . By using the Lebesgue dominated convergence theorem, the compactness of implies that we obtain . This gives the approximate controllability.

4. Conclusions

An approximate controllability result for nonlinear Fractional Sobolev type stochastic differential equations driven by mixed fractional Brownian motion is obtained by means of contraction principle fixed point theorems under the compactness assumption. It is also proven that the approximate controllability of linear deterministic system implies the approximate controllability of nonlinear Fractional Sobolev type stochastic differential equations driven by mixed fractional Brownian motion in Hilbert spaces.

References

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