Polynomial iISS for a class of Timoshenko equations with external disturbances and infinite history memory

This paper investigates the robustness of Timoshenko equations subject to external disturbances and infinite history memory within the integral input-to-state stability (iISS) framework.

While polynomial iISS (PiISS) is a powerful tool for characterizing the influence of these two factors, the existing definitions of PiISS rely on graph norms of the state, which are inconsistent with the classical iISS notion for infinite-dimensional systems.

In this paper, we provide a rigorous PiISS definition for a class of Timoshenko equations solely via the state norm, and derive sufficient conditions for PiISS under the Tolksdorf condition on the relaxation function, as well as exponential iISS (EiISS) under a stronger condition.

The well-posedness is established by using the semigroup and elliptic equation theories.

The PiISS and EiISS are assessed by constructing appropriate Lyapunov functionals and employing various a priori estimates of solutions, thereby fully characterizing the impact of disturbances and history memory on system stability.