Controllability for semilinear SPDEs with dynamic boundary conditions and gradient nonlinearities

Extend controllability results for semilinear forward stochastic parabolic equations with Dirichlet boundary conditions and globally Lipschitz semilinear terms to equations with dynamic boundary conditions whose semilinear terms depend on the state and its spatial gradients both in the bulk and on the boundary, for broad classes of nonlinearities beyond globally Lipschitz where compactness embeddings are unavailable.

Background

The paper establishes controllability for linear forward stochastic parabolic equations with dynamic boundary conditions. For semilinear SPDEs with Dirichlet boundaries and globally Lipschitz nonlinearities, global null-controllability results have been obtained in prior work.

The authors note that moving to dynamic boundary conditions with gradient-dependent nonlinearities introduces substantial technical obstacles, notably the absence of compactness embeddings in stochastic settings, which hinders standard approaches used in deterministic and Dirichlet cases.

They highlight that for broader nonlinearity classes discussed in related deterministic literature, this extension remains unresolved.