  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) x (2+2) hyperbolic PIDE-ODE system. Then backstepping is applied
to obtain a control law guaranteeing closed-loop stability of the origin in the
H^1 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.
