Optimal chattering solutions for longitudinal vibrations of a nonhomogeneous bar

TL;DR

This paper investigates the control of longitudinal vibrations in a nonhomogeneous bar, demonstrating that optimal control involves an infinite number of switchings, known as chattering control, to minimize mean square deviation.

Contribution

It proves that the optimal control for the vibration minimization problem is a chattering control with infinitely many switchings in finite time.

Findings

Optimal control exhibits chattering behavior.

Infinite switchings occur in finite time.

Chattering control minimizes mean square deviation.

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.