Stabilization and controllability of first-order integro-differential hyperbolic equations

TL;DR

This paper investigates the stabilization and controllability of first-order linear integro-differential hyperbolic equations, establishing their equivalence and using Fredholm transformations and the method of moments to analyze controllability criteria.

Contribution

It introduces a Fredholm transformation approach to link stabilization with controllability for these equations and reduces controllability to Fattorini's criterion in specific cases.

Findings

Finite-time stabilization is equivalent to exact controllability.

Fredholm transformation maps the system to a finite-time stable target.

Controllability reduces to Fattorini's criterion in particular cases.

Abstract

In the present article we study the stabilization of first-order linear integro-differential hyperbolic equations. For such equations we prove that the stabilization in finite time is equivalent to the exact controllability property. The proof relies on a Fredholm transformation that maps the original system into a finite-time stable target system. The controllability assumption is used to prove the invertibility of such a transformation. Finally, using the method of moments, we show in a particular case that the controllability is reduced to the criterion of Fattorini.