ISS with Respect to Boundary and In-domain Disturbances for a Coupled Beam-String System

TL;DR

This paper investigates the input-to-state stability of a coupled beam-string PDE system under boundary and in-domain disturbances, providing ISS properties without unbounded operator complications.

Contribution

It introduces a Lyapunov functional approach for ISS analysis of coupled PDEs, avoiding unbounded operators and deriving explicit ISS gains.

Findings

Established ISS properties for the coupled beam-string system.

Derived explicit ISS gains without involving derivatives of disturbances.

Ensured well-posedness and regularity conditions for solutions.

Abstract

This paper addresses the robust stability of a boundary controlled system coupling two partial differential equations (PDEs), namely beam and string equations, in the presence of boundary and in-domain disturbances under the framework of input-to-state stability (ISS) theory. Well-posedness assessment is first carried out to determine the regularity of the disturbances required for guaranteeing the unique existence of the solution to the considered problem. Then, the method of Lyapunov functionals is applied in stability analysis, which results in the establishment of some ISS properties {}{with respect to} disturbances. As the analysis is based on the \emph{a priori} estimates of the solution to the PDEs, it allows avoiding the invocation of unbounded operators while obtaining the ISS gains in their original expression without involving the derivatives of boundary disturbances.