Exact controllability of stochastic differential equations with multiplicative noise

TL;DR

This paper establishes the exact controllability of certain stochastic differential equations with multiplicative noise under the Kalman rank condition, and applies the results to stochastic heat equations.

Contribution

It proves exact controllability for stochastic differential equations with multiplicative noise satisfying the Kalman condition, extending controllability theory to stochastic systems.

Findings

Exact controllability of stochastic controlled equations proven.

Controllability results apply to stochastic heat equations.

The Kalman rank condition is key for controllability.

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.