Singularity of Optimal Control for a Timoshenko Beam
We study singularities of optimal solutions in a problem of controlling the Timoshenko beam vibrations. The Timoshenko beam vibrations are described by a system of two coupled hyperbolic equations. Controls are introduced as external bounded forces. We consider the problem of minimizing the mean square deviation of the Timoshenko beam from the equilibrium position. For some initial conditions we reduce this problem to the optimal control problem for ordinary differential equations. We study the case of two-dimensional controls. For some initial positions of the beam, we prove that the optimal solutions have a Fuller type singularity. We give an asymptotic representation of the corresponding family of optimal trajectories.