Rapid Stabilization of Timoshenko Beam by PDE Backstepping

In this paper, we present rapid boundary stabilization of a Timoshenko beam with anti-damping and anti-stiffness at the uncontrolled boundary, by using infinite-dimensional backstepping. We introduce Riemann transformation to map states into set coordinates that verify 1-D hyperbolic PIDE-ODE system. Then backstepping is applied obtain control law guaranteeing closed-loop stability origin in $L^{2}$ sense. Arbitrarily can be achieved adjusting parameters, has not been previous results. Finally, numerical simulation shows effectiveness proposed controller. This result extends work which considered slender Kelvin-Voigt damping, allowing destabilizing conditions attaining an arbitrarily convergence rate.