  This paper considers the stability problem of a linear time invariant system
in feedback with a string equation. A new Lyapunov functional candidate is
proposed based on the use of augmented states which enriches and encompasses
the classical Lyapunov functional proposed in 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 together with Legendre polynomials. Numerical examples illustrate
the potential of our approach through three scenari: a stable ODE perturbed by
the PDE, an unstable open-loop ODE stabilized by the PDE and an unstable
closed-loop ODE stabilized by the PDE.
