Uniformly exponentially stable approximations for Timoshenko beams

In this note, Timoshenko beams with interior damping and boundary damping are stud-ied from the viewpoints of control theory and numerical approximation. Especially, the uniform exponential stabilities of the beams are studied. The meaning of uniform ex-ponential stability in this paper is two-fold: The first one is in the classical sense and also is concisely called exponential stability by many authors; The second one is that the semi-discretization systems, which are derived from an exponentially stable continuous beam by some semi-discretization schemes, are uniformly exponentially stable with re-spect to the discretized parameter. To investigate uniform exponential stability of contin-uous and discrete systems, five completely different methods, which are stability theory of port-Hamiltonian system, direct method of Lyapunov functional, perturbation theory of C0-semigroup, spectral analysis of unbounded operator and frequency standard of expo-nential stability for contractive semigroup, are involved. Especially, a new method, which is based on the frequency domain characteristics of uniform exponential stability of C0- semigroup of contractions, is established to verify the uniform exponential stability of semi-discretization systems derived from coupled system. The effectiveness of the numer-ical approximating algorithms is verified by numerical simulations. (c) 2023 Elsevier Inc. All rights reserved.

Automated summary

This paper studies Timoshenko beams with interior damping and boundary damping from the viewpoints of control theory and numerical approximation. The uniform exponential stability of the beams is particularly investigated. The meaning of uniform exponential stability in this paper refers to both the classical sense and the stability of semi-discretization systems with respect to the discretized parameter. Various methods, including stability theory, Lyapunov functional method, perturbation theory, spectral analysis, and frequency standard, are involved to investigate the uniform exponential stability of continuous and discrete systems. A new method based on frequency domain characteristics is proposed to verify the uniform exponential stability of semi-discretization systems derived from coupled systems. Numerical simulations demonstrate the effectiveness of the numerical approximating algorithms.