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Approximate controllability of retarded semilinear stochastic system with non local conditions. (English) Zbl 1330.34115

The approximate controllability in a finite time interval for semilinear infinite-dimensional time-varying stochastic systems with nonlocal conditions is considered. It is generally assumed that the values of admissible controls are unconstrained. First of all, using the theory of infinite-dimensional linear operators and the integral form of the stochastic differential state equation, the linear controllability operator is defined and discussed, which is used to construct a suitable admissible control. Next, a nonlinear contraction operator defined in appropriate spaces is introduced in order to prove the existence of a solution for the semilinear state equation.
The main result of the paper is a sufficient condition for approximate controllability in a given time interval proved by using the well-known Banach fixed-point theorem. Many remarks and comments on controllability problems for stochastic and semilinear control systems are also given. The special case of finite-dimensional semilinear stochastic systems is also discussed. Finally, two examples which illustrate theoretical considerations are presented. It should be pointed out that similar controllability problem for stochastic systems with delays in control has been considered in the paper by J. Klamka [Bull. Pol. Acad. Sci., Tech. Sci. 55, No. 1, 23–29 (2007; Zbl 1203.93190)].

MSC:

34K35 Control problems for functional-differential equations
34K30 Functional-differential equations in abstract spaces
93C25 Control/observation systems in abstract spaces
34K50 Stochastic functional-differential equations

Citations:

Zbl 1203.93190
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References:

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