ISS of a Clamped-Free Damped String for the Configurations Associated with the Loss of the Riesz-Spectral Properties

This paper deals with the Input-to-State Stability (ISS) of a clamped-free damped string with respect to boundary disturbances for the configurations associated with the loss of the Riesz-spectral properties. Specifically, for most of the values of the physical parameters (namely the stiffness parameter and the damping coefficient), the ISS property of the clamped free damped string can be established based on the fact that the underlying disturbance-free operator is a Riesz-spectral operator. However, such a Riesz-spectral property does not hold true for certain configurations of the physical parameters of the damped string. This paper specifically investigates the establishment of an ISS estimate for these configurations. The proposed strategy relies on the projection of the original system trajectories in a Riesz basis obtained by adequately completing the set of eigenvectors of the disturbance-free operator.