- Open Access
Approximate controllability of impulsive neutral stochastic differentialequations with fractional Brownian motion in a Hilbert space
Advances in Difference Equations volume 2014, Article number: 113 (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]).
The purpose of this paper is to investigate the approximate controllability problemfor the class of impulsive neutral stochastic functional differential equations withfinite delay and fractional Brownian motion in a Hilbert space of the form
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 ).
Moreover, has the following Wiener integral representation:
where is a Wiener process, and is the kernel given by
for , where and
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.
We recall that for their scalar product in ζ is given by
Let us consider the operator from ζ to defined by
Moreover, for any , we have
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.
We define the infinite-dimensional fBm on Y with covariance Q as
where are real, independent fBm’s. TheY-valued process is Gaussian, starts from 0, has mean zero and covariance:
In order to define Wiener integrals with respect to the Q-fBm, we introducethe space of all Q-Hilbert Schmidt operators. We recall that is called a Q-Hilbert-Schmidt operator, if
and that the space equipped with the inner product is a separable Hilbert space.
Let ; be a function with values in , the Wiener integral of ϕ with respectto is defined by
where is the standard Brownian motion.
Lemma 2.1 (see )
Ifsatisfiesthen the above sum in (2.1) is well defined as X-valued random variable and we have
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.
For every there exists such that
Now, we present the mild solution of the problem (1.1):
Definition 3.1 An X-valued process is called a mild solution of equation (1.1) if
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.
(H3) The function g is -valued, and there exist constants, , , such that for all and the following inequalities are satisfied:
(H4) The function is continuous in the quadratic mean sense: for all, the equality
(H5) The function satisfies .
(H6) The functions are continuous and there exist finite positiveconstants , , such that for all and the inequalities and are valid.
In order to study the approximate controllability for the system (1.1), we introducethe following linear differential system:
The controllability operator associated with (3.2) is defined by
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.
Lemma 3.3 For any there exists such that
Now for anyand, we define the control function in the followingform:
Lemma 3.4 There exists a positive real constantsuch that, for all, we have
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 .
is a closed subset of provided with the norm . For any , consider the operator on defined as follows:
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.
Let , where , and let be sufficiently small. Then for any fixed, it follows from Holder’s inequality and theassumptions on the theorem that
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.
Step 2: Now, we are going to show that is a contraction mapping in with some to be specified latter. Let , we obtain for any fixed
By the Lipschitz property of and f combined with Holder’sinequality, we obtain
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.
Proof Let be a fixed point of . By using the stochastic Fubini theorem, it caneasily be seen that
It follows from the assumption on f and g that there exists such that
for all . Then there is a subsequence still denoted by which converges weakly to, say, in .
From the above equation, we have
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.
Let us consider the following stochastic control partial neutral functionaldifferential equation with finite variable delays driven by a fractional Brownianmotion:
where is a fractional Brownian motion and are continuous functions.
To study this system, let and let the operator be given by with
It is well known that A is the infinitesimal generator of an analyticsemigroup on X. Furthermore, A has discretespectrum with eigenvalues , and the corresponding normalized eigenfunctions aregiven by
In addition is a complete orthonormal basis in X and
for and . It follows from this representation that is compact for every and that for every .
In order to define the operator , we choose a sequence , set , and assume that
Define the fractional Brownian motion in Y by
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 .
Bainov D, Simeonov P: Impulsive Differential Equations: Periodic Solutions andApplications. Longman, Harlow; 1993.
Rogovchenko YV: Nonlinear impulsive evolution systems and applications to populationmodels. J. Math. Anal. Appl. 1997, 207: 300–315. 10.1006/jmaa.1997.5245
Benchohra M, Henderson J, Ntouyas SK: Existence results for impulsive multivalued semilinear neutral functionalinclusions in Banach spaces. J. Math. Anal. Appl. 2001, 263: 763–780. 10.1006/jmaa.2001.7663
Benchohra M, Henderson J, Ntouyas SK: Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, New York; 2006.
Bogdan P, Marculescu R: Towards a science of cyber-physical systems design. In Proc. 2nd ACM/IEEE Int’l Conf. Cyber-Physical Systems. IEEE Comput. Soc., Los Alamitos; 2011:99–108.
Marculescu R, Bogdan P: Cyberphysical systems: workload modeling and design optimization. IEEE Des. Test Comput. 2011, 28(4):78–87. July-August
Bogdan P: Implantable pacemakers control and optimization via fractional calculusapproaches: a cyber-physical systems perspective. Proceedings of the IEEE/ACM Third International Conference on Cyber-PhysicalSystems (ICCPS’12) 2012.
Balachandran K, Sakthivel R: Controllability of integrodifferential systems in Banach spaces. Appl. Math. Comput. 2001, 118: 63–71. 10.1016/S0096-3003(00)00040-0
Balachandran K, Sakthivel R: Controllability of functional semilinear integrodifferential systems inBanach spaces. J. Math. Anal. Appl. 2001, 225: 447–457.
Fu X: Controllability of abstract neutral functional differential systems withunbounded delay. Appl. Math. Comput. 2004, 151: 299–314. 10.1016/S0096-3003(03)00342-4
Ahmed HM: Boundary controllability of nonlinear fractional integrodifferentialsystems. Adv. Differ. Equ. 2010., 2010: Article ID 279493
Klamka J: Stochastic controllability of linear systems with delay in control. Bull. Pol. Acad. Sci., Tech. Sci. 2007, 55(1):23–29.
Klamka J: Stochastic controllability of linear systems with state delays. Int. J. Appl. Math. Comput. Sci. 2007, 17: 5–13.
Klamka J: Stochastic controllability of systems with variable delay in control. Bull. Pol. Acad. Sci., Tech. Sci. 2008, 56: 279–284.
Klamka J: Stochastic controllability and minimum energy control of systems withmultiple delays in control. Appl. Math. Comput. 2008, 206(2):704–715. 10.1016/j.amc.2008.08.059
Klamka J: Stochastic controllability of systems with multiple delays in control. Int. J. Appl. Math. Comput. Sci. 2009, 19(1):39–47.
Ahmed HM: Controllability of fractional stochastic delay equations. Lobachevskii J. Math. 2009, 30: 195–202. 10.1134/S1995080209030019
Sakthivel R, Ganesh R, Suganya S: Approximate controllability of fractional neutral stochastic system withinfinite delay. Rep. Math. Phys. 2012, 70: 291–311. 10.1016/S0034-4877(12)60047-0
Sakthivel R, Ganesh R, Ren Y, Anthoni SM: Approximate controllability of nonlinear fractional dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 2013, 18: 3498–3508. 10.1016/j.cnsns.2013.05.015
Sakthivel R, Ren Y: Approximate controllability of fractional differential equations withstate-dependent delay. Results Math. 2013, 63: 949–963. 10.1007/s00025-012-0245-y
Ren Y, Bi Q, Sakthivel R: Stochastic functional differential equations with infinite delay driven by G -Brownian motion. Math. Methods Appl. Sci. 2013, 36: 1746–1759. 10.1002/mma.2720
Ren Y, Dai H, Sakthivel R: Approximate controllability of stochastic differential system driven by aLevy process. Int. J. Control 2013, 86: 158–1164.
Sakthivel R, Suganya S, Anthoni SM: Approximate controllability of fractional stochastic evolution equations. Comput. Math. Appl. 2012, 63: 660–668. 10.1016/j.camwa.2011.11.024
Sakthivel R: Complete controllability of stochastic evolution equations with jumps. Rep. Math. Phys. 2011, 68: 163–174. 10.1016/S0034-4877(12)60003-2
Mandelbrot BB, Van Ness JW: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 1968, 10: 422–437. 10.1137/1010093
Boufoussi B, Hajji S: Neutral stochastic functional differential equations driven by a fractionalBrownian motion in a Hilbert space. Stat. Probab. Lett. 2012, 82: 1549–1558. 10.1016/j.spl.2012.04.013
Pazy A Applied Mathematical Sciences 44. In Semigroups of Linear Operators and Applications to Partial DifferentialEquations. Springer, New York; 1983.
Sakthivel R: Approximate controllability of impulsive stochastic evolution equations. Funkc. Ekvacioj 2009, 52: 381–393. 10.1619/fesi.52.381
I am very grateful to the referees and the Editor in Chief for their carefulreading and helpful comments.
The author declares that he has no competing interests.
About this article
Cite this article
Ahmed, H.M. Approximate controllability of impulsive neutral stochastic differentialequations with fractional Brownian motion in a Hilbert space. Adv Differ Equ 2014, 113 (2014). https://doi.org/10.1186/1687-1847-2014-113
- fractional Brownian motion
- approximate controllability
- mild solution
- impulsive neutral stochastic functional differential equations
- fixed point theorem