New Global Carleman Estimates and Null Controllability for Forward/Backward Semilinear Parabolic SPDEs

In this paper, we study the null controllability for parabolic SPDEs involving both the state and the gradient of the state. To start with, an improved global Carleman estimate for linear forward (resp. backward) parabolic SPDEs with general random coefficients and square-integrable source terms is derived. Based on this, we further develop a new global Carleman estimate for linear forward (resp. backward) parabolic SPDEs with source terms in the Sobolev space of negative order, which enables us to deal with the global null controllability for linear backward (resp. forward) parabolic SPDEs with gradient terms. As a byproduct, a special weighted energy-type estimate for the controlled system that explicitly depends on the parameters $λ,μ$ and the weighted function $θ$ is obtained, which makes it possible to extend the previous linear null controllability to semi-linear backward (resp. forward) parabolic SPDEs by applying the fixed-point argument in an appropriate Banach space.