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Balasubramaniam, P.; Muthukumar, P.

Approximate controllability of second-order stochastic distributed implicit functional differential systems with infinite delay. (English) Zbl 1176.93013

J. Optim. Theory Appl. 143, No. 2, 225-244 (2009).
Summary: In this paper, sufficient conditions for the approximate controllability of the following stochastic semilinear abstract functional differential equations with infinite delay are established \[ \begin{aligned} d\bigl[x^{\prime}(t)-g(t,x_{t})\bigr]&=\bigl[Ax(t)+f(t,x_{t})+Bu(t)\bigr]dt+G(t,x_{t})dW(t),\quad \text{a.e on}\;t\in J:=[0,b],\\x_{0}&=\varphi\in {\mathfrak{B}},\\x^{\prime}(0)&=\psi \in H,\end{aligned} \] where the state \(x(t)\in H,x_{t}\) belongs to phase space \({\mathfrak{B}}\) and the control \(u(t)\in L^{\mathcal F}_2 (J,U)\), 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.

Cited in 21 Documents

MSC:

93B05 Controllability
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
93C25 Control/observation systems in abstract spaces
93C20 Control/observation systems governed by partial differential equations

Keywords:

controllability of systems; cosine functions operators; distributed control systems; stochastic partial functional differential equations
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\textit{P. Balasubramaniam} and \textit{P. Muthukumar}, J. Optim. Theory Appl. 143, No. 2, 225--244 (2009; Zbl 1176.93013)
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References:

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