Controllability of semilinear stochastic evolution equations in Hilbert space.

*(English)*Zbl 1031.93040Semilinear, stochastic evolution equations defined in Hilbert space are considered. It is assumed that the differential state equation contains both linear and nonlinear parts and that the values of controls are unconstrained. Using the Banach fixed-point theorem and semigroup theory, a sufficient condition for exact complete controllability in a given time interval is formulated and proved. In the proof of the main result, methods of nonlinear functional analysis are used extensively. Moreover, as an illustrative example, the complete exact controllability of a semilinear stochastic partial differential equation is discussed. Finally, several remarks, comments and relations to controllability results existing in the literature are given. It should be mentioned that similar stochastic controllability problems have been recently investigated in the paper reviewed above [R. Subramaniam and K. Balanchandran, Korean Comput. Appl. Math. 9, 538-589 (2002; Zbl 1031.93039)].

Reviewer: J.Klamka (Katowice)

##### MSC:

93B05 | Controllability |

93C25 | Control/observation systems in abstract spaces |

93C10 | Nonlinear systems in control theory |

93E03 | Stochastic systems in control theory (general) |

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |