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

TL;DR

This paper proves that the heat equation with general dynamic boundary conditions can be controlled to zero from the boundary within any positive time, using Carleman estimates.

Contribution

It establishes boundary null controllability for a heat equation with broad dynamic boundary conditions, extending previous results to more general boundary dynamics.

Findings

Boundary null controllability holds for all positive times.

Carleman estimates are used to prove controllability.

Controllability applies to initial data in L^2 spaces.

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 $\partial_tu - \gamma\Delta u = 0$ in $\Omega\times(0,T)$ with the non-homogeneous general dynamic boundary conditions $\partial_tu_{\Gamma} -\delta\Delta_\Gamma u_{\Gamma}+ \gamma\partial_{\nu}u + \beta u_{\Gamma} = g$ on $\Gamma\times(0,T)$ is always null controllable from the boundary for every $T>0$ and initial data $(u_0,u_{\Gamma,0})\in L^2(\Omega)\times L^2(\Gamma)$.