Exact boundary controllability and stabilizability of a degenerated Timoshenko beam
Günter Leugering, Yue Wang, Qiong Zhang
TL;DR
The paper addresses exact boundary controllability and exponential stabilization of a degenerate Timoshenko beam, where shear and bending stiffnesses vanish at one boundary. It develops a rigorous framework of weighted Sobolev spaces and bilinear forms to handle weak and strong degeneracy, and uses multiplier identities together with the Hilbert Uniqueness Method to obtain observability and controllability results under Dirichlet, Robin, and Neumann conditions. The results generalize boundary-control theory to non-uniform, damaged beams and establish conditions under which boundary feedback yields exponential energy decay. Collectively, the work advances control strategies for degenerate elastic systems with practical relevance to structural damping and vibration suppression.
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 $μ_a\in [0,2)$ for $a\in\{K,EI\}$, where $K(x)$ denotes shear stiffness and $EI(x)$ bending stiffness. We differentiate between weak degeneracy $μ_a\in [0,1)$ and strong degeneracy $μ_a\in [1,2)$, 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.