Exact controllability of stochastic differential equations with multiplicative noise

Abstract

One proves that the $n$-D stochastic controlled equation $dX+AXdt=\sigma (X)dW+Budt$, where $σ\in\mbox{Lip}((\R^n,Ł(\R^d,\R^n))$ and the pair $A\in Ł({\mathbb{R}}^n)$, $B\in Ł({\mathbb{R}}^m,{\mathbb{R}}^n)$ satisfies the Kalman rank condition, is exactly controllable in each $y\in {\mathbb{R}}^n$, $\sigma (y)=0$ on each finite interval $(0,T)$. An application to approximate controllability to stochastic heat equation is given.