Controllability And Observability Of Coupled Stochastic Parabolic Systems

This dissertation is devoted to the controllability of coupled linear stochastic parabolic systems by one control force and observability of the systems through only an observer.Controllability and observability are two fundamental problems in the modern control theory.Controllability is mainly concerned with feasibility problems.In order to establish the null controllability of coupled stochastic parabolic systems,the Lebeau-Robbiano technique is adopted in this dissertation.The key point is to prove an observability estimate for some backward stochastic coupled parabolic system,when terminal values belong to a finite dimensional space.Meanwhile,a counterexample is given to explain that different from deterministic systems,the Kalman rank condition for the null controllability of coupled stochastic parabolic systems does not hold any longer.On the other hand,an observability inequality and unique continuation property for general linear coupled systems governed by several stochastic parabolic equations through one observer are derived.The Carleman estimates method is adopted.The results show that under certain conditions,the local information on one component of solution vectors can determine the whole information of solutions.Compared to the known results,the observability result of this dissertation has not limitations on the number of equations,and requirements on regularity of some coefficients are also relaxed.