Lack of controllability of thermal systems with memory

TL;DR

This paper demonstrates that systems with heat equations and memory, especially with Laplacian outside the memory term, generally lack controllability to zero, contrasting with classical heat equations.

Contribution

It proves that lack of controllability is a universal property for systems with smooth memory kernels, extending previous specific examples to all such systems.

Findings

Lack of controllability holds for all systems with smooth memory kernels.

Controllability properties of heat equations with memory differ from classical heat equations.

Counterexamples show some initial conditions cannot be controlled to zero.

Abstract

Heat equations with memory of Gurtin-Pipkin type have controllability properties which strongly resemble those of the wave equation. Instead, recent counterexamples show that when the laplacian appears also out of the memory term, the control properties do not parallel those of the (memoryless) heat equation, in the sense that there are $L^2$-initial conditions which cannot be controlled to zero. The proof of this fact (presented in previous papers) consists in the construction of two quite special examples of systems with memory which cannot be controlled to zero. Here we prove that lack of controllability holds in general, for every systems with smooth memory kernel.