Controllability of semilinear stochastic systems in Hilbert spaces.

*(English)*Zbl 1109.93305The author studies weak approximate and complete controllability of the following semilinear stochastic system

$$\begin{array}{c}dx\left(t\right)=\left[Ax\right(t)+Bu(t)+F(t,x\left(t\right),u\left(t\right)\left)\right]\phantom{\rule{0.166667em}{0ex}}dt+{\Sigma}(t,x(t),u(t\left)\right)dw\left(t\right),\\ x\left(0\right)={x}_{0},\phantom{\rule{1.em}{0ex}}t\in [0,T]\xb7\end{array}\phantom{\rule{2.em}{0ex}}\left(1\right)$$

where $A:H\to H$ is an infinitesimal generator of strongly continuous semigroup $S(\xb7)$, $B\in L(0,H)$, $F:[0,T]\times H\times U\to H$, ${\Sigma}:[0,T]\times H\times U\to {L}_{2}^{0}$; $H,U$ are Hilbert spaces and $w$ is a suitably chosen Wiener process. To this end the author introduces the weak approximate controllability concept for stochastic systems which is a weaker concept than the usual ones, that is, approximate controllability and complete controllability. The author presents sufficient conditions for weak approximate and complete controllability of (1) and provides the paper with several examples.

Reviewer: Messoud A. Efendiev (Berlin)

##### MSC:

93B05 | Controllability |

60H15 | Stochastic partial differential equations |

60J65 | Brownian motion |

93E03 | General theory of stochastic systems |