Boundary Average Controllability Of Stochastic Heat Equations

Some progress has been made in the study of average controllability of control systems,such as the average controllability of stochastic heat equation,the average controllability of stochastic Schr（?）dinger equation,the average con-trollability of fractional ordinary diferential equation and fractional difusion equation,etc.But as far as the author knows,most of them focus on the internal situation of control.Because of the need of theory and practice,it is a meaningful work to study the average controllability of the boundary when some control can only be applied to the boundary.The research on this aspect is still in its infancy.In this paper,we study the average controllability of boundary control for stochastic heat equations with arbitrarily distributed difusion coeffcients.First,in solving the existence of weak solutions,we associate the solution of boundary control system with the solution of abstract Cauchy problem.From the existence of the solution of the abstract Cauchy problem,it is obtained that the solution of the boundary control system also exists.For the uncer-tainty of the distribution of random variableα,we use a simple function to approximate its probability density function.At the same time,the solution of the dual system is decomposed by Fourier,so that the observability esti-mation of feature sequence_{e_j}_j=1^∞can be used in the subsequent derivation.Next,we divide the time set E into infnite connected subsets,and prove the estimation formula that holds on these subsets.Then,through the iterative method,the estimators on the subset are accumulated to obtain that the sys-tem is averagely observable.Finally,we derive the average null controllability and average approximate controllability of the system from the average ob-servability of the system through the variational method.In this paper,the problem of average controllability of stochastic heat equation with control on the interior is extended to the case of control on the boundary,and the average null controllability and average approximate controllability of stochastic heat equation with boundary control are obtained.