CONTROLLABILITY OF SOME COUPLED STOCHASTIC PARABOLIC SYSTEMS WITH FRACTIONAL ORDER SPATIAL DIFFERENTIAL OPERATORS BY ONE CONTROL IN THE DRIFT

摘要

This paper is addressed to a study of the null/approximate controllability for a class of coupled systems governed by two linear forward stochastic parabolic equations with fractional order spatial differential operators. Our method is based on the Lebeau-Robbiano strategy. The key is to establish a suitable observability estimate for some coupled fractional order backward stochastic parabolic systems with terminal states in finite dimensional spaces. Compared to deterministic coupled parabolic systems, the coupling appearing in diffusion terms in the stochastic case introduces quite interesting new phenomena. We present a somewhat surprising counterexample to show that the controllability of coupled stochastic parabolic systems is not robust with respect to the coupling coefficient in diffusion terms. This indicates that the usual Carleman-type estimate approach does not seem to work for our controllability problem. Moreover, our controllability results for parabolic systems with fractional order spatial differential operators through one control are new, even when the considered- system degenerates to a deterministic one.