A Lyapunov approach for the exponential stability of a damped Timoshenko beam

TL;DR

This paper uses a Lyapunov approach within the port-Hamiltonian framework to prove exponential stability of a damped Timoshenko beam with spatially varying parameters, providing explicit decay rates and numerical validation.

Contribution

It introduces a Lyapunov-based method within port-Hamiltonian framework to establish exponential stability with explicit decay rates for a damped Timoshenko beam.

Findings

Proved existence of solutions for the damped Timoshenko beam system.

Established exponential stability with an explicit decay rate.

Validated results through numerical simulations.

Abstract

In this technical note, we consider the stability properties of a viscously damped Timoshenko beam equation with spatially varying parameters. With the help of the port-Hamiltonian framework, we first prove the existence of solutions and show, by an appropriate Lyapunov function, that the system is exponentially stable and has an explicit decay rate. The explicit exponential bound is computed for an illustrative example of which we provide some numerical simulations.