Optimal chattering solutions for longitudinal vibrations of a nonhomogeneous bar

Abstract

We consider a control problem for longitudinal vibrations of a nonhomogeneous bar with clamped ends. The vibrations of the bar are controlled by an external force which is distributed along the bar. For the minimization problem of mean square deviation of the bar we prove that the optimal control has an infinite number of switchings in a finite time interval, i.e., the optimal control is the chattering control.