Boundary null controllability of the heat equation with Wentzell boundary condition and Dirichlet control

TL;DR

This paper proves that the heat equation with Wentzell boundary conditions can be controlled to zero using boundary controls, employing spectral analysis, moment method, and penalized HUM to validate controllability and approximate controls.

Contribution

It establishes boundary null controllability for the heat equation with Wentzell boundary conditions and develops a penalized HUM method for control approximation.

Findings

Proves null controllability at any positive time T.

Reduces controllability to a moment problem using spectral analysis.

Validates controllability results with a penalized HUM approach.

Abstract

We consider the linear heat equation with a Wentzell-type boundary condition and a Dirichlet control. Such a boundary condition can be reformulated as one of dynamic type. First, we formulate the boundary controllability problem of the system within the framework of boundary control systems, proving its well-posedness. Then we reduce the question to a moment problem. Using the spectral analysis of the associated Sturm-Liouville problem and the moment method, we establish the null controllability of the system at any positive time $T$. Finally, we approximate minimum energy controls by a penalized HUM approach. This allows us to validate the theoretical controllability results obtained by the moment method.