Exact controllability of stochastic differential equations with multiplicative noise

Abstract

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