Carleman Estimates for Backward Anisotropic Stochastic Parabolic Equations with General Dynamic Boundary Conditions and Applications

Abstract

We investigate a backward anisotropic stochastic parabolic equation with general dynamic boundary conditions, where the drift involves both $\mathbb{L}^2$ and $\mathbb{H}^{-1}$ bulk--surface terms. We first establish the well-posedness of this equation. Subsequently, we derive a new Carleman estimate through a two-step approach. In the first step, using a weighted identity method together with a careful treatment of the boundary integral terms arising from the dynamic boundary conditions, we obtain an intermediate Carleman estimate for backward anisotropic stochastic parabolic equations without weak divergence source terms. In the second step, a duality method combined with suitable optimization techniques is employed to incorporate the weak divergence source terms. As applications of the derived Carleman estimate, we address two control problems. First, we establish null controllability for forward anisotropic stochastic parabolic equations with general dynamic boundary conditions. These equations involve both reaction and convection terms, with adapted, bounded stochastic bulk--surface coefficients. Moreover, we provide an explicit estimate of the null controllability cost, i.e., a bound on the minimal norm of controls required to drive the system to zero at the terminal time $T$. Second, we study an insensitizing control problem for this class of equations. The goal is to determine controls for systems with partially unknown initial data such that a given energy functional remains insensitive to small perturbations of these data. In this work, the functional involves the norm of the state over a localized bulk--surface region, together with the norm of its tangential gradient over a localized boundary region.