Controllability and lack of controllability with smooth controls in viscoelasticity via moment methods

TL;DR

This paper investigates the controllability of linear viscoelastic systems with memory, showing that smoother controls can reach smoother targets in the absence of memory, but memory acts as an obstacle to full controllability extension.

Contribution

It extends controllability results from wave equations to viscoelastic systems with memory, highlighting the limitations imposed by memory effects.

Findings

Smoother controls can reach smoother targets in memoryless systems.

Memory introduces obstructions to full controllability extensions.

Partial controllability results are established for systems with memory.

Abstract

In this paper we study controllability of a linear equation with persistent memory when the control belongs to $ H^k_0(0,T;L^2(\ZOMq)) $. In the case the memory is zero, our equation is reduced to the wave equation and a result due to Everdoza and Zuazua informally states that smoother targets can be reached by using smoother controls. In this paper we prove that this result can be partially extended to systems with memory, but that the memory is an obstruction to a complete extensions.