Controllability of a linear system with persistent memory via boundary traction

TL;DR

This paper proves that the controllability of a linear viscoelastic system with memory via boundary traction can be inferred from the controllability of its purely elastic counterpart, extending known results to more delicate boundary conditions.

Contribution

It establishes that controllability of the elastic system implies controllability of the viscoelastic system with boundary traction, despite the complex nature of the associated functional sequences.

Findings

Viscoelastic system controllability follows from elastic system controllability.

The proof handles the complexity of Riesz-Fisher sequences in boundary traction control.

Extends controllability results to systems with memory effects.

Abstract

We consider a linear viscoelastic system of Maxwell-Boltzmann type. Hence, viscosity contributes a memory term to the elastic equation. The system is controlled via the traction exerted on a part $\Gamma_1$ of the boundary of the body. We prove that if the associated elastic system (i.e. the elastic system without memory) is exactly controllable then the viscoelastic system is exactly controllable too. This is similar to the known result when the boundary deformation is controlled, but the proof is far more delicate since controllability under boundary traction corresponds to the fact that a certain sequence of functions is a Riesz-Fisher sequence, but not a Riesz sequence.