Arbitrary Decay Rate for Euler-Bernoulli Beam by Backstepping Boundary Feedback

We consider a problem of stabilization of the Euler-Bernoulli beam. The beam is controlled at one end (using position and moment actuators) and has the ldquoslidingrdquo boundary condition at the opposite end. We design the controllers that achieve any prescribed decay rate of the closed loop system, removing a long-standing limitation of classical ldquoboundary damperrdquo controllers. The idea of the control design is to use the well-known representation of the Euler-Bernoulli beam model through the Schrodinger equation, and then adapt recently developed backstepping designs for the latter in order to stabilize the beam. We derive the explicit integral transformation (and its inverse) of the closed-loop system into an exponentially stable target system. The transformation is of a novel Volterra/Fredholm type. The design is illustrated with simulations.