Backstepping-Based Exponential Stabilization of Timoshenko Beam with Prescribed Decay Rate

This is an open access article under the CC BY-NC-ND license.In this paper, we present a rapid boundary stabilization of a Timoshenko beam with anti-damping and anti-stiffness at the uncontrolled boundary, by using PDE backstepping. We introduce a transformation to map the Timoshenko beam states into a (2+2) × (2+2) hyperbolic PIDE-ODE system. Then backstepping is applied to obtain a control law guaranteeing closed-loop stability of the origin in the H1 sense. Arbitrarily rapid stabilization can be achieved by adjusting control parameters. Finally, a numerical simulation shows that the proposed controller can rapidly stabilize the Timoshenko beam. This result extends a previous work which considered a slender Timoshenko beam with Kelvin-Voigt damping, allowing destabilizing boundary conditions at the uncontrolled boundary and attaining an arbitrarily rapid convergence rate

WELL-POSEDNESS AND ASYMPTOTIC STABILITY OF SOLUTIONS TO A BRESSE SYSTEM WITH TIME VARYING DELAY TERMS AND INFINITE MEMORIES

We consider the Bresse system in bounded domain with timevarying delay terms in the internal feedbacks and innite memories acting in the three equations of the system. First, we prove the well-posedness of its solutions in Sobolev spaces by means of semigroup theory. Furthermore, the asymptotic stability is also discussed by using an appropriate Lyapunovfunctional.We consider the Bresse system in bounded domain with timevarying delay terms in the internal feedbacks and innite memories actingin the three equations of the system. First, we prove the well-posedness ofits solutions in Sobolev spaces by means of semigroup theory. Furthermore,the asymptotic stability is also discussed by using an appropriate Lyapunovfunctional