Boundary feedback stabilization by piecewise constant time delay for the wave equation

TL;DR

This paper investigates the stabilization of a vibrating string using boundary feedback with specific constant and piecewise constant delays, demonstrating exponential stability under certain conditions.

Contribution

It introduces conditions under which boundary feedback with piecewise constant delays stabilizes a vibrating string, extending previous results on delay effects.

Findings

Exponential stability achieved with delays of 4L/c, 8L/c, 12L/c ...

Stability also holds with piecewise constant delays at 4L/c and 8L/c

Delays of certain values preserve the stabilizing effect of feedback.

Abstract

For vibrating systems, a delay in the application of a feedback control can destroy the stabilizing effect of the control. In this paper we consider a vibrating string that is fixed at one end and stabilized with a boundary feedback with delay at the other end. We show that for certain feedback parameters the system is exponentially stable with constant delays of the form 4L/c, 8L/c, 12L/c ... Moreover, we show that the system is exponentially stable with piecewise constant delays that attain the values 4L/c and 8L/c.