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

Author

Listed:
  • P. Balasubramaniam

    (Gandhigram Rural University)

  • P. Muthukumar

    (Gandhigram Rural University)

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 2 ℱ (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.

Suggested Citation

  • P. Balasubramaniam & P. Muthukumar, 2009. "Approximate Controllability of Second-Order Stochastic Distributed Implicit Functional Differential Systems with Infinite Delay," Journal of Optimization Theory and Applications, Springer, vol. 143(2), pages 225-244, November.
    • Handle: RePEc:spr:joptap:v:143:y:2009:i:2:d:10.1007_s10957-009-9564-x
    DOI: 10.1007/s10957-009-9564-x
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    Cited by:

    1. Ren, Yong & Hou, Tingting & Sakthivel, R. & Cheng, Xing, 2014. "A note on the second-order non-autonomous neutral stochastic evolution equations with infinite delay under Carathéodory conditions," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 658-665.
    2. Y. Ren & D. D. Sun, 2010. "Second-order Neutral Stochastic Evolution Equations with Infinite Delay under Carathéodory Conditions," Journal of Optimization Theory and Applications, Springer, vol. 147(3), pages 569-582, December.

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