Boundary null controllability for a heat equation with general dynamical boundary condition

Abstract

Let $\Omega \subset {\mathbb{R}}^N$ be a bounded open set with Lipschitz continuous boundary $\Gamma$. Let $\gamma >0$, $\delta \ge 0$ be real numbers and $\beta$ a nonnegative measurable function in $L^{\infty }(\Gamma )$. Using some suitable Carleman estimates, we show that the linear heat equation ${\partial }_{t}u-\gamma \Delta u=0$ in $\Omega \times (0,T)$ with the non-homogeneous general dynamic boundary conditions ${\partial }_t{u}_{\Gamma }-\delta {\Delta }_{\Gamma }{u}_{\Gamma }+\gamma {\partial }_{\nu }u+\beta u_{\Gamma }=g$ on $\Gamma \times (0,T)$ is always null controllable from the boundary for every $T>0$ and initial data $(u_0,u_{\Gamma ,0})\in L^2(\Omega )\times L^2(\Gamma )$.