Backstepping-based Rapid Stabilization of Two-layer Timoshenko Composite Beams

In this paper, we investigate the rapid stabilization of two-layer Timoshenko composite beams with anti-damping and anti-stiffness at the uncontrolled boundaries. This work extends our previous result on single-layer Timoshenko beams. While the problem of stabilization for two-layer composite beams has been previously studied, the obtention of an arbitrarily fast decay rate is a novel result, as well as considering anti-damping and anti-stiffness in the boundaries (which can possibly lead to rapid divergence). Our approach is based on the introduction of a Riemann transformation of the states of two-layer Timoshenko beams into a 1-D hyperbolic PIDE (partial integro-differential equations)-ODE system. Then, PDE backstepping is used to design a control law resulting in closed-loop stability of the origin in the L2 sense. An arbitrarily rapid convergence rate can be obtained by adjusting control parameters. Copyright (c) 2023 The Authors.