Carleman Estimates and Controllability of Forward Stochastic Parabolic Equations with General Dynamic Boundary Conditions

Abstract: We derive a new Carleman estimate for a general backward stochastic parabolic equation with dynamic boundary conditions, incorporating weak divergence source terms in both the bulk and surface equations. This estimate is obtained through two main steps: first, by refining a known Carleman estimate for backward stochastic parabolic equations to explicitly account for the dependence of the parameters on the final control time (T); second, by applying a duality technique to address weak divergence source terms. As an application, we prove the null and approximate controllability of forward stochastic parabolic equations with dynamic boundary conditions, which involve both reaction and convection terms with bounded adapted coefficients, as well as general second-order parabolic operators. Additionally, we provide an explicit estimate for the null-controllability cost in terms of the final control time (T) and the coefficients of the equation.