Singular solutions for vibration control problems

Abstract

Optimal control problem for the system of partial differential equations of hyperbolic type is considered. By using the Fourier method this problem is reduced to the optimal control problem for the corresponding Fourier coefficients. For some special initial data we prove the existence of optimal solutions with a countable number of switchings on a finite time interval and optimal spiral-like solutions which attain the origin in a finite time making a countable number of rotations. The problem of controlling the vibrations of the Timoshenko beam is considered as an example of the optimal control problem for linear system of PDE.