CONTROLLABILITY AND OBSERVABILITY OF SOME COUPLED STOCHASTIC PARABOLIC SYSTEMS

This paper is devoted to a study of controllability and observability problems for some stochastic coupled linear parabolic systems only by one control and through an observer, respectively. In order to get a null controllability result, the Lebeau-Robbiano technique is adopted. The key point is to prove an observability inequality for certain stochastic coupled backward parabolic system by an iteration, when terminal values belong to a finite dimensional space. Different from deterministic systems, Kalman-type rank conditions for the controllability of stochastic coupled parabolic systems do not hold any more. Meanwhile, based on the Carleman estimates method, an observability inequality and unique continuation property for general stochastic linear coupled parabolic systems through an observer are derived.