On Controllability of a Non-Homogeneous Elastic Sstring with Memory

TL;DR

This paper investigates the controllability of a non-homogeneous elastic string with memory, reducing it to a moment problem and establishing a Riesz basis property, with implications for control and observation in such systems.

Contribution

It introduces a novel reduction of controllability to a moment problem and proves a Riesz basis property for a related family of functions, advancing control theory for systems with memory.

Findings

Controllability is linked to a non-standard moment problem.

A Riesz basis property is established for a family of functions close to nonharmonic exponentials.

The results have implications for observability of dual systems.

Abstract

We are motivated by the problem of control for a non-homogeneous elastic string with memory. We reduce the problem of controllability to a non-standard moment problem. The solution of the latter problem is based on an auxiliary Riesz basis property result for a family of functions quadratically close to the nonharmonic exponentials. This result requires the detailed analysis of an integro-differential equation and is of interest in itself for Function Theory. Controllability of the string implies observability of a dual system.