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

Abstract

Let $Ω\subset\mathbb R^N$ be a bounded open set with Lipschitz continuous boundary $Γ$. Let $γ>0$, $δ\ge 0$ be real numbers and $β$ a nonnegative measurable function in $L^\infty(Γ)$. Using some suitable Carleman estimates, we show that the linear heat equation $\partial_tu - γΔu = 0$ in $Ω\times(0,T)$ with the non-homogeneous general dynamic boundary conditions $\partial_tu_Γ -δΔ_Γu_Γ+ γ\partial_νu + βu_Γ = g$ on $Γ\times(0,T)$ is always null controllable from the boundary for every $T>0$ and initial data $(u_0,u_{Γ,0})\in L^2(Ω)\times L^2(Γ)$.