In conditions for the approximate controllability for a class

In this paper, we formulate a set of sufficient conditions for the approximate controllability for a class of second-order stochastic non-autonomous integrodifferential inclusions in Hilbert space. We establish the results with the help of resolvent operators and Bohnenblust-Karlin’s fixed point theorem is to prove the main result. An application is given to illustrate the main result.end{abstract}{f Keywords:} Approximate controllability, Stochastic integrodifferential inclusions, Resolvent operators, Evolution equations, Mild solutions \{f 2010 Mathematics Subject Classification:} 34A08, 34K40, 34G20. section{Introduction}
oindentpar Differential inclusions have wide applications in engineering, economics and in optimal control theory. Many authors studied the existence, controllability and stability of differential inclusions cite{p3,p7,p8,p25,p34,p35,p36}. Controllability is one of the elementray concepts in mathematical control theory, which plays a vital role in both engineering and sciences. Controllability generally means that it is possible to steer dynamical control systems from an arbitrary initial state to an arbitrary final state using the set of admissible controls. There are two basic theories of controllability can be identified as approximate controllability and exact controllability. Most of the criteria, which can be met in the literature are formulated for finite dimensional system. But in the infinite dimensional system, many unsolved problems are still exist as for as controllability is concerned. In the case of infinite dimensional system, controllability can be distinguished as approximate and exact controllability. Approximate controllability means that the system can be governed to arbitrary small neighborhood of final state whereas exact controllability allows to govern the system to arbitrary final state. In otherwords the approximate controllability gives the possibility of governing the system of states which forms a dense subspace in the state space. Many authors studied the existence, controllability and approximate controllability of the stochastic differential equations and inclusions in the first and second order cite{p8,p9,p13,p19,p21,p23,p24,p25,p28,p29,p30,p31,p33,p37,p38}. par  Stochastic Differential Equations(SDE’s) naturally refer to the time dynamics of the evolution of a state vector, based on the (approximate) physics of the real system, together with a driving noise process. The noise process can be assumed in several ways. It often symbolizes processes not included in the model, but present in the real system. Random differential and evolution systems play a crucial role in characterizing many social, physical, biological, medical and engineering problems cite{p6,p10,p26,p27,p32} and references therein. par In the past years, the authors cite{p1,p2,p4,p12,p16,p17}, investigated the existence of abstract second order initial value problem for the non-autonomous system. In cite{p14,p15}, studied the analytic resolvent operators for the integral equations in Banach space and he also studied the resolvent operator for integral equations in Banach space. In cite{p18} Henr’ studied the existence of solutions of a second order abstract functional cauchy problem with nonlocal conditions and in cite{p20} he also studied the existence of solutions of non-autonomous abstract cauchy problem of second order integrodifferential equations by the use of resolvent operators instead of cosine family of operators. In cite{p16} Henr’iquez, investigated the existence of solutions of non-autonomous second order functional differential equations with infinite delay by using Leray Schauder alternative fixed point theorem. The approximate controllability of second order non autonomous integrodifferential inclusions by resolvent operators have not studied yet. Motivated by the above facts, we establish the sufficient conditions for the approximate controllability of second order stochastic integrodifferetial inclusions by resolvent operators by using Bohenblust-Karlin fixed point theorem.par In this paper, we establish the set of sufficient conditions for the approximate controllability for a non-autonomous second order stochastic integrodifferential inclusions in Hilbert space of the form,egin{eqnarray}  frac{d}{dt}x'(t) &in A(t) x(t)+ int_{0}^{t}G(t,s)x(s)ds+B u(t) + Sigma(t,x(t))dw(t), t in J: =0,b, label{rn1}\  x(0) &=& y, x'(0)=z  label{rn2}end{eqnarray}where $A(t) : D(A(t)) subseteq X
ightarrow X$ is a closed linear operator, $G(t,s): D(G) subseteq X
ightarrow X$ is a linear operator and the control function the state $x(cdot)$ takes the values in the separable real Hilbert space $X$; Further $Sigma: J imes X
ightarrow  mathcal{L}_{2}^{0}(mathbb{K},X)$ is nonempty, bounded, closed and convex multivalued map  and the control function $u(cdot)$ is given in $mathcal{L}(J, U)$, a Hilbert space of admissible control functions with $U$ as Hilbert space. $B$ is a bounded linear operator from $U$ into $X$.par The paper is organised as follows: In section 2, we present some basic notations and preliminaries  and in section 3, we studied the approximate controllability results by resolvent operator and in section 4, an application is provide to illustrate the main results.