Finite-time stabilization and impulse control of heat equation with dynamic boundary conditions

TL;DR

This paper demonstrates that a multi-dimensional heat equation with dynamic boundary conditions can be stabilized in finite time using impulse controls, providing explicit decay estimates and employing advanced spectral and boundary analysis techniques.

Contribution

It introduces a novel approach to impulse controllability for heat equations with dynamic boundary conditions, including explicit decay estimates and a sophisticated analytical framework.

Findings

System is impulse null controllable at any positive time

Provides explicit exponential decay estimates

Employs a combination of logarithmic convexity and spectral analysis

Abstract

In this paper, we study the impulse controllability of a multi-dimensional heat equation with dynamic boundary conditions in a bounded smooth domain. Using a recent approach based on finite-time stabilization, we show that the system is impulse null controllable at any positive time via impulse controls supported in a nonempty open subset of the physical domain. Furthermore, we infer an explicit estimate for the exponential decay of the solution. The proof of the main result combines a logarithmic convexity estimate and some spectral properties associated to dynamic boundary conditions. In our setting, the nature of the equations, which couple intern-boundary phenomena, makes it necessary to go into quite sophisticated estimates incorporating several boundary terms.