Lyapunov Stability Analysis of a String Equation Coupled With an Ordinary Differential System

This paper considers the stability problem of a linear time invariant system in feedback with a string equation. A new Lyapunov functional is proposed using augmented states that enriches and encompasses the classical functionals of the literature. It results in tractable stability conditions expressed in terms of linear matrix inequalities. This methodology follows from the application of the Bessel inequality to the projections over the Legendre polynomials. Numerical examples illustrate the potential of our approach through three scenario: a stable ODE perturbed by the PDE, an unstable open-loop ODE, and an unstable closed-loop ODE stabilized by the PDE.