Approximate controllability of impulsive neutral stochastic differentialequations with fractional Brownian motion in a Hilbert space
© Ahmed; licensee Springer. 2014
Received: 23 December 2013
Accepted: 8 April 2014
Published: 2 May 2014
Approximate controllability for impulsive neutral stochastic functionaldifferential equations with finite delay and fractional Brownian motion in aHilbert space are studied. The results are obtained by using semigroup theory,stochastic analysis, and Banach’s fixed point theorem. Finally, an exampleis given to illustrate the application of our result.
MSC: 60H15, 60G22, 93B05, 34A37.
The impulsive differential systems are used to describe processes which are subjectedto abrupt changes at certain moments. The impulsive effects exist widely in thedifferent areas of the real world such as mechanics, electronics,telecommunications, neural networks, finance and economics, etc. (see [1–7]). On the other hand, it is well known that the stochastic control theoryis a stochastic generalization of classical control theory. As one of thefundamental concepts in mathematical control theory, controllability plays animportant role both in deterministic and stochastic control theory. Controllabilitygenerally means that it is possible to steer a dynamical control system from anarbitrary initial state to an arbitrary final state using the set of admissiblecontrols (see [8–17]). Moreover, the approximate controllability means that the system can besteered to arbitrary small neighborhood of final state. Approximate controllablesystems are more prevalent and very often approximate controllability is completelyadequate in applications (see [18–24]).
where A is the infinitesimal generator of an analytic semigroup of boundedlinear operators, , in a Hilbert space X, is a fractional Brownian motion on a real andseparable Hilbert space Y, the initial data and the control function is given in , the Hilbert space of admissible control functionswith U a Hilbert space. The symbol B stands for a bounded linearfrom U into X. The functions () are continuous, , where and represent the right and left limits of at , respectively, and , , are appropriate Lipschitz type functions. Here be the Banach space of all continuous functionsξ from into , equipped with the supremum norm.
2 Fractional Brownian motion
Fix a time interval and let be a complete probability space.
Suppose that is the one-dimensional fractional Brownian motionwith Hurst parameter . That is, is a centered Gaussian process with covariancefunction (see ).
We put if .
We will denote by ζ the reproducing kernel Hilbert space of the fBm. Infact ζ is the closure of set of indicator functions with respect to the scalar product.
The mapping can be extended to an isometry from ζonto the first Wiener chaos and we will denote by the image of φ under this isometry.
Let X and Y be two real, separable Hilbert spaces and let be the space of bounded linear operators fromY to X. For the sake of convenience, we shall use the samenotation to denote the norms in X, Y and . Let be an operator defined by with finite trace , where () are non-negative real numbers and () is a complete orthonormal basis in Y.
and that the space equipped with the inner product is a separable Hilbert space.
where is the standard Brownian motion.
Lemma 2.1 (see )
3 Approximate controllability
Let be the infinitesimal generator of an analyticsemigroup, , of bounded linear operators on X. It iswell known that there exist and such that for every .
If is uniformly bounded and analytic semigroup such that, where is the resolvent set of A, then it ispossible to define the fractional power for , as a closed linear operator on its domain. Furthermore, the subspace is dense in X and the expression defines a norm in . If represents the space endowed with the norm , then the following properties are well known.
Lemma 3.1 ()
Let , then is a Banach space.
If , then the injection is continuous.
- (3)For every there exists such that
Now, we present the mild solution of the problem (1.1):
- (iii)for arbitrary , we have(3.1)
In this paper, we will make the following assumptions.
(H1) The operator A is the infinitesimal generator of an analytic semigroup,, consisting of bounded linear operators onX. Furthermore, there exist constants M and such that for every the inequalities and hold.
(H2) There exist finite positive constants , , such that the function satisfies the following Lipschitz conditions: for all and the inequalities and are valid.
(H5) The function satisfies .
(H6) The functions are continuous and there exist finite positiveconstants , , such that for all and the inequalities and are valid.
where and denote the adjoint of B and S,respectively.
Let be the state value of (1.1) at terminal stateT, corresponding to the control u and the initial valueφ. Denote by the reachable set of system (1.1) at terminal timeT, its closure in X is denoted by .
Definition 3.2 The system (1.1) is said to be approximately controllable onthe interval if .
Lemma 3.2 (see )
The linear control system (3.2) is approximately controllable onif and only ifstrongly as.
Proof The proof of this lemma similar to the proof of the Lemma 2.5(see ). □
Theorem 3.1 Assume assumptions (H1)-(H6) are satisfied. Then, forall, the system (1.1) has a mild solutionon.
Proof Fix and let us consider .
It will be shown that, for all , the operator has a fixed point. This fixed point is then asolution of equation (1.1). To prove this result, we divide the subsequent proofinto two steps.
Step 1: For arbitrary , let us prove that is continuous on the interval in the -sense.
Hence using the strong continuity of 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 continuous from the right in. A similar argument shows that it is also continuousfrom the left in . Thus is continuous on in the -sense.
Then there exists such that and is a contraction mapping on and therefore has a unique fixed point, which is amild solution of equation (1.1) on . This procedure can be repeated in order to extendthe solution to the entire interval in finitely many steps. This completes theproof. □
Theorem 3.2 Assume that (H1)-(H6) are satisfied. Further, if thefunctions f and g are uniformly bounded, andis compact, then the system (1.1) is approximatelycontrollable on.
for all . Then there is a subsequence still denoted by which converges weakly to, say, in .
On the other hand, by Lemma 3.2, the operator strongly as for all , and, moreover, . Thus, by the Lebesgue dominated convergence theoremthe compactness of implies that as . This gives the approximate controllability of(1.1). □
In this section, we present an example to illustrate our main result.
where is a fractional Brownian motion and are continuous functions.
for and . It follows from this representation that is compact for every and that for every .
where and is a sequence of one-dimensional fractional Brownianmotions mutually independent.
Define , , and . Define the bounded operator by , , . Therefore, with the above choice, the system (4.1)can be written into the abstract form (1.1) and all conditions of Theorem 3.2are satisfied. Thus by Theorem 3.2, the stochastic partial neutral functionaldifferential equation with finite variable delays driven by a fractional Brownianmotion is approximately controllable on .
I am very grateful to the referees and the Editor in Chief for their carefulreading and helpful comments.
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