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Controllability of semilinear stochastic systems. (English) Zbl 1097.93034
The authors present sufficient conditions for approximate and complete controllability for a class of semi linear stochastic systems. The line of reasoning is based on the Piccard approximation used to prove a fixed point property of the nonlinear operator which represents a solution of the system. The approximate controllability is proved for the systems for which the associated linear system is controllable (for linear stochastic systems the complete and approximate controllabilities are equivalent [N. I. Mahmudov and A. Denker, Int. J. Control 73, No. 2, 144–151 (2000; Zbl 1031.93033)]). To prove the complete controllability of the semilinear stochastic system the authors assume additionally that the system matrix (of a linear part of the system) is nonnegative and selfadjoint and the input matrix multiplied by its transposition is positive.
MSC:
93E03General theory of stochastic systems
93B05Controllability
47H10Fixed point theorems for nonlinear operators on topological linear spaces
93B18Linearizability of systems
37H10Generation, random and stochastic difference and differential equations