1. Introduction

The aim main of this paper is to study the principle concepts of the approximate controllability for complicated classes of fractional order 0<α<1 Riemann-Liouville type stochastic perturbed control systems driven by mixed fractional Brownian motion The following form is the system under our consideration,

where, the Riemann-Liouville fractional derivative of order 0<α<1. −A is the infinitesimal generator of a compact analytic semigroup of uniformly bounded linear operators is a bounded linear operator in a real separable Hilbert space X. Assume that . The space is a Hilbert such that , equipped with norm .

is a continuous at } with the norm . is F₀–measurable X- valued random variable independent of W and which defined on a complete probability space The control function is a Hilbert space and the operator B from U into X is a bounded linear operator such that there exists constant is a standard cylindrical Brownian motion defined on with values in a Hilbert space K. 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 is a Q-fractional Brownian motion with Hurst index defined in a complete probability space with values in a 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 . The functions are continuous functions.

Approximate controllability of stochastic control system driven by fractional Brownian motion has been interested by many authors; Sakthivel ^[19] 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 and wiener process in real separable Hilbert space.

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 order Riemann-Liouville type stochastic perturbed control systems driven by mixed fractional Brownian motion in a real separable Hilbert spaces.

The rest of this paper is organized as follows, in section 2, we will introduced some concepts, definitions and some lemmas of semigroup theory and 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

(1)

•  If , then the increments of B^H are non-correlated, and consequently independent. So B^H 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

(2)

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

(3)

(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

(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

(6)

•  If , we have

(7)

(8)

Lemma (2.2), ^[6]:

For any functions Φ, , we have

i)

ii)

From this Lemma above, we obtain

(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

(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

(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

(12)

Definition (2.2), ^[18]:

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

(13)

where,

Definition (2.4), ^[18]:

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

where, n-1< α <n.

Lemma (2.6), ^[17]:

Let -A be the infinitesimal generator of an analytic semigroup S(t), on a Hilbert space X. If -∆A is a bounded linear operator on X then –(A+∆A) is the infinitesimal generator of an analytic semigroup on X.

Remark (2.3):

Assume that is a compact analytic semigroup of uniformly bounded operators in X , that is , there exists M > 1 such that .

Definition (2.5):

An -valued process x(t) is called a mild solution of the system (1) if and, for satisfies the integral equation

(15)

where,

is a Mainardi's function.

Lemma (2.7):

If is a compact analytic semigroup then the family of operators have the following properties:

i. For any fixed , the operator is a linear and bounded, i.e. for any , there exists such that

ii. For any , there exists such that

iii. For any where,

iv. is a strongly continuous, which mean that for every and then if .

v. The operator is a compact operator in X for t > 0.

3. Main Result of the Approximately Controllable

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

a) The operator is a compact for any t > 0.

b) The linear fractional order system of corresponding the system (3.41) which has following form:

(16)

is an approximately controllable on [0,T].

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

Also, F is a uniformly bounded. In other word, there exists such that

d) The function satisfies, for every t ∈ [0, T]

and there exists such that .

e) is F_t – adapted with respect to t, such that, for every satisfy the following:

i. Exists.

ii.

iii. There exists , such that

where,

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,

Now,

The controllability operator associated with control system (16) is defined by

(17)

Also, for any θ > 0 and , the operator is defined by

(18)

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

Lemma (3.1), ^[12]:

The linear fractional order deterministic system in (1) is an approximately controllable on [0, T] if and only if the operator

Lemma (3.2), ^[13]: For any , there exists and , such that

(19)

where, ,

(20)

Lemma (3.3): There exists positive real constant , such that for all

(21)

where,

Proof

Let and be a fixed. From the equation (21), we have:

Applying Holder’s inequality and by using Ito isometry , Lemmas (2.5), (2.7) and the assumptions (a)-(e), we obtain

where,

Now,

For any θ > 0, consider the operator on defined as follows:

(22)

Also, for any , the subset of is define as

Lemma (3.4):

, There exists .

Proof:

To prove that there exists in other word,

Suppose that this is not true, then for each > 0, there exists such that , for , t may depending upon . However, on the other hand, we have

Applying Holder’s inequality and by using Ito isometry , Lemmas (2.5), (2.7), (3.3) and the assumptions (a)-(e), we obtain

Hence,

Therefore,

By dividing both sides of above inequality by and taking the limit as →∞, which is a contradiction. Thus, for each , there exists positive number such that .

Now,

Let where,

(23)

(24)

Lemma (3.5):

Assume that the assumptions (a) – (e) hold, then for any , and for any , for

Proof

Let and , we have

Applying Holder’s inequality and by using Ito isometry , Lemmas (2.5), (2.7), (3.3) and the assumptions (a)-(e), we obtain

which mean , then

Lemma (3.6):

For any the operator is a contraction on , provided that

Proof:

Let and , we have

By using lemma (2.7) and assumption (c), we get

By taking the supremum over for both sides, we have

Therefore,

Where, , hence is a contraction.

Lemma (3.7):

Assume that the assumptions (a) – (e) hold, then the operator maps bounded sets into bounded sets in .

Proof:

Let , for we have

Applying Holder’s inequality and by using Ito isometry , Lemmas (2.5), (2.7), (3.3) and the assumptions (a)-(e), we obtain

where,

By taking the supremum over for both sides, we get

Therefore, for each , we get . Then maps bounded sets into bounded sets in .

Lemma (3.8):

Assume that the assumptions (a) – (e) hold, then is a continuous on .

Proof:

Let be a sequence in such that as in . For each , we have

From Ito isometry and lemma (2.7), we obtain

Therefore, it follows from the continuity of and that for each and , using the Lebesgue dominated convergence theorem that for all , we conclude as Implying that Hence, is continuous on .

Lemma (3.9):

If the assumptions (a) – (e) are hold, then for , the set is equicontinuous.

Proof:

Let such that . Then, from the equation (24), we have

Now, from Lemma (2.7), noting the fact that for every there exists such that, whenever , for every Therefore, when we have

The right hand of the inequality above tends to 0 as and . Hence for , the set is equicontinuous.

Lemma (3.10):

If the assumption (a) is hold. Then for each the set is relatively compact in .

Proof:

Let be a fixed and for every, we define

(25)

Then, from the definition of semigroup we can easily be written in the form, from the equality (25), we have

Then, from the compactness of , we obtain that the set is relatively compact in X for every such that Moreover, for , we can easily prove that is convergent to in , as and , hence the set is relatively compact in .

Lemma (3.11):

If the assumptions (a) – (e) are hold. Then is a completely continuous.

Proof:

From lemma (3.7) and (3.9), for the operator is uniformly bounded and the set is equicontinuous by Appling the Arzela –Ascoli theorem, it results that for , the set is relatively compact.

we obtain that is a completely continuous.

Theorem (3.1):

If the assumptions (a)-(e) are satisfied. Then for each the control system (1) has a mild solution on [0, T], provided that .

Proof:

For any and for any ,, for By using lemma (3.6) and (3.11) with applying Krasnoselskii's fixed point theorem, we conclude that the operator has a fixed point, which gives rise to mild solution of system (1) with stochastic control function given in (20). This completes the proof.

Theorem (3.2):

If the assumptions (a) – (e) are satisfied. Then the stochastic control system (1) is approximately controllable on [0,T].

Proof:

For every let be a fixed point of the operator in the space which is a mild solution under the stochastic control function in (20) of the stochastic control system (1). Then, we have

(25)

By using the assumptions (c) the function F is uniformly bounded, such that , for any . Then, for all there is a subsequence of sequence denoted by which is weakly converging to say F(s) in X. Also, there is a subsequence of sequence denoted by which is weakly converging to say in . On the other hand, from the assumption (b), for all the operator Strongly as and .

By using the Lebesgue dominated convergence theorem, we obtain as . This gives the approximate controllability.

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