Null Controllability for Wave Equations with Memory

TL;DR

This paper establishes the null controllability of wave equations with memory terms, ensuring the system reaches equilibrium with the memory component also vanishing, using a moving control support and duality methods.

Contribution

It introduces a novel controllability result for wave equations with memory, utilizing a moving geometric control condition and duality to handle degenerate coupled PDE-ODE systems.

Findings

Memory-type null controllability is achieved under Moving Geometric Control Condition.

Controllability is proved via an observability inequality with moving observation sets.

The approach reduces the problem to classical null controllability for a coupled PDE-ODE system.

Abstract

We study the memory-type null controllability property for wave equations involving memory terms. The goal is not only to drive the displacement and the velocity (of the considered wave) to rest at some time-instant but also to require the memory term to vanish at the same time, ensuring that the whole process reaches the equilibrium. This memory-type null controllability problem can be reduced to the classical null controllability property for a coupled PDE-ODE system. The later is viewed as a degenerate system of wave equations, the velocity of propagation for the ODE component vanishing. This fact requires the support of the control to move to ensure the memory-type null controllability to hold, under the so-called Moving Geometric Control Condition. The control result is proved by duality by means of an observability inequality which employs measurements that are done on a moving observation open subset of the domain where the waves propagate.