Null controllability of a volume-surface reaction-diffusion equation with dynamic boundary conditions

This article focuses on investigating the null controllability of a volume-surface reaction-diffusion equation with dynamic boundary conditions. Notably, the reaction and diffusion coefficients depend on the state variable in the bulk and on its surface. Our main result establishes the local null controllability of the quasi-linear equation under certain conditions on the regularity of diffusion and reaction coefficients as well as initial data. To this end, we approach the problem by first addressing the question of null controllability in the framework of a inhomogeneous linearized equation. Next, we derive new estimates of both control and state, allowing us to apply a local inversion theorem to obtain the local null controllability of the quasi-linear equation.