Exact Controllability of Semilinear Stochastic Evolution Equation

Abstract

In this paper we study the exact controllability of the following semilinear stochastic evolution equation in a Hilbert space $X$

$$ dx(t)=\{Ax(t)+Bu(t)+f(t,\omega,x(t),u(t)) \}dt + \{\Sigma(t) +\sigma(t,\omega,x(t),u(t)) \}dw(t), $$

where the control $u$ is a stochastic process in the Hilbert space $U$, $A:D(A)\subset X\rightarrow X,$ is the infinitesimal generator of a strongly continuous semigroup $\left\{S(t)\right\}_{t\geq 0}$ on $X$ and $B\in L(U,X)$. To this end, we give necessary and sufficient conditions for the exact controllability of the linear part of this system

$$ dx(t)=Ax(t)dt+Bu(t)dt+\Sigma(t)dw(t). $$

Then, under a Lipschitzian condition on the non linear terms $f$ and $\sigma$ we prove that the exact controllability of this linear system is preserved by the semilinear stochastic system. Moreover, we obtain explicit formulas for a control steering the system from the initial state $\xi_0$ to a final state $\xi_1$ on time $T >0$, for both system, the linear and the nonlinear one. Finally, we apply this result to a semilinear damped stochastic wave equation.