Boundary controllability and source reconstruction in a viscoelastic string under external traction

TL;DR

This paper investigates boundary controllability and source reconstruction in a viscoelastic string affected by external traction, extending moment methods to time-dependent coefficients for inverse problem applications.

Contribution

It introduces an extension of moment methods to control a viscoelastic string with time-varying coefficients due to external traction, addressing a gap in control theory.

Findings

Extended moment methods to time-dependent coefficients

Achieved controllability results for viscoelastic strings under external traction

Provided a framework for inverse source identification

Abstract

Treatises on vibrations devote large space to study the dynamical behavior of an elastic system subject to known external tractions. In fact, usually a "system" is not an isolated body but it is part of a chain of mechanisms which disturb the "system" for example due to the periodic rotation of shafts. This kind of problem has been rarely studied in control theory. In the specific case we shall study, the case of a viscoelastic string, the effect of such external action is on the horizontal component of the traction, and so it affects the coefficients of the corresponding wave type equation, which will be time dependent. The usual methods used in controllability are not naturally adapted to this case. For example at first sight it might seem that moment methods can only be used in case of coefficients which are constant in time. Instead, we shall see that moment methods can be extended to study controllability of a viscoelastic string subject to external traction and in particular we shall study a controllability problem which is encountered in the solution of the inverse problem consisting in the identification of a distributed disturbance source.