Mean Square Finite-Approximate Controllability of Semilinear Stochastic Differential Equations with Non-Lipschitz Coefficients

In this paper, we present a study on mean square approximate controllability and finite-dimensional mean exact controllability for the system governed by linear/semilinear infinite-dimensional stochastic evolution equations. We introduce a stochastic resolvent-like operator and, using this operator, we formulate a criterion for mean square finite-approximate controllability of linear stochastic evolution systems. A control is also found that provides finite-dimensional mean exact controllability in addition to the requirement of approximate mean square controllability. Under the assumption of approximate mean square controllability of the associated linear stochastic system, we obtain sufficient conditions for the mean square finite-approximate controllability of a semilinear stochastic systems with non-Lipschitz drift and diffusion coefficients using the Picard-type iterations. An application to stochastic heat conduction equations is considered.