Approximate Controllability of Second-Order Stochastic Distributed Implicit Functional Differential Systems with Infinite Delay |
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Authors: | P Balasubramaniam P Muthukumar |
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Institution: | 1. Department of Mathematics, Gandhigram Rural University, Gandhigram, 624 302, Tamil Nadu, India
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Abstract: | In this paper, sufficient conditions for the approximate controllability of the following stochastic semilinear abstract functional differential equations with infinite delay are established $$\begin{array}{@{}l@{}}d\biglx^{\prime}(t)-g(t,x_{t})\bigr]=\biglAx(t)+f(t,x_{t})+Bu(t)\bigr]dt+G(t,x_{t})dW(t),\\\noalign{\vskip3pt}\quad \mbox{a.e on}\ t\in J:=0,b],\\\noalign{\vskip3pt}x_{0}=\varphi\in {\mathfrak{B}},\\\noalign{\vskip3pt}x^{\prime}(0)=\psi \in H,\end{array}$$ where the state x(t)∈H,x t belongs to phase space ${\mathfrak{B}}$ and the control u(t)∈L 2 ? (J,U), in which H,U are separable Hilbert spaces and d is the stochastic differentiation. The results are worked out based on the comparison of the associated linear systems. An application to the stochastic nonlinear wave equation with infinite delay is given. |
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