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Approximate Controllability of Second-Order Stochastic Distributed Implicit Functional Differential Systems with Infinite Delay

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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

$$\begin{array}{@{}l@{}}d\bigl[x^{\prime}(t)-g(t,x_{t})\bigr]=\bigl[Ax(t)+f(t,x_{t})+Bu(t)\bigr]dt+G(t,x_{t})dW(t),\\\noalign{\vskip3pt}\quad \mbox{a.e on}\ t\in J:=[0,b],\\\noalign{\vskip3pt}x_{0}=\varphi\in {\mathfrak{B}},\\\noalign{\vskip3pt}x^{\prime}(0)=\psi \in H,\end{array}$$

where the state x(t)∈H,x t belongs to phase space \({\mathfrak{B}}\) and the control u(t)∈L ℱ2 (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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Authors and Affiliations

  1. Department of Mathematics, Gandhigram Rural University, Gandhigram, 624 302, Tamil Nadu, India

    P. Balasubramaniam & P. Muthukumar

Authors
  1. P. Balasubramaniam
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  2. P. Muthukumar
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Correspondence to P. Balasubramaniam.

Additional information

Communicated by F. Zirilli.

This work was supported by NBHM Project 48/1/2007-R&D-II/7446, Govt. of India. The authors are grateful to Professor F. Zirilli and two anonymous referees for valuable suggestions improving this paper.

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Balasubramaniam, P., Muthukumar, P. Approximate Controllability of Second-Order Stochastic Distributed Implicit Functional Differential Systems with Infinite Delay. J Optim Theory Appl 143, 225–244 (2009). https://doi.org/10.1007/s10957-009-9564-x

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