Exact boundary controllability and stabilizability of a degenerated Timoshenko beam

TL;DR

This paper studies how to control and stabilize a degenerate Timoshenko beam at its boundary, establishing observability inequalities and feedback controls that ensure exponential stabilization, extending previous wave equation results to this more complex system.

Contribution

It provides new controllability and stabilization results for degenerate Timoshenko beams, addressing different degeneracy levels and boundary conditions, with methods applicable to real-world structural control.

Findings

Established observability inequalities for degenerate Timoshenko equations.

Derived conditions for exact boundary controllability using HUM.

Proved exponential stabilization with boundary feedback controls.

Abstract

This paper investigates the boundary controllability and stabilizability of a Timoshenko beam subject to degeneracy at one end, while control is applied at the opposite boundary. Degeneracy in this context is measured by the real parameters for $\mu_a\in [0,2)$ for $a\in\{K,EI\}$, where $K(x)$ denotes shear stiffness and $EI(x)$ bending stiffness. We differentiate between weak degeneracy $\mu_a\in [0,1)$ and strong degeneracy $\mu_a\in [1,2)$, which may occur independently in shear and bending. Our study establishes observability inequalities for both weakly and strongly degenerate equations under Dirichlet, Robin, and Neumann boundary conditions. Using energy multiplier techniques and the Hilbert Uniqueness Method (HUM), we derive conditions for exact boundary controllability and show that appropriate boundary state and velocity feedback controls at the non-degenerate end can stabilize the system exponentially. Extending results previously obtained for the 1-dimensional wave equation in \cite{AlabauCannarsaLeugering2017}, this study highlights new control strategies and stabilization effects specific to the degenerate Timoshenko beam system, addressing challenges pertinent to real-world structural damping and control applications.