A new approach for stability analysis of 1-D wave equation with time delay

TL;DR

This paper introduces a novel spectral analysis method for the stability of 1-D wave equations with time delay, establishing conditions on feedback gain and delay for exponential stability, including cases with irrational delays.

Contribution

It presents a new spectral analysis approach and comprehensive stability conditions for wave equations with time delay, covering positive and irrational delays, and analyzes stability region behavior.

Findings

Exponential stability if and only if the delay is an even number.

Derived stability regions for different even multiples of delay.

Analyzed the impact of delay perturbations on robustness.

Abstract

In our manuscript, we develop a new approach for stability analysis of one-dimensional wave equation with time delay. The major contribution of our work is to develop a new method for spectral analysis. We derive sufficient and necessary conditions for the feedback gain and time delay which guarantee the exponential stability of the closed-loop system. Comparing with similar conditions developed in the past literatures, we discuss all the situation when the time delay is positive, including when it is irrational. We prove that the exponential stability can be achieved if and only if the time delay is an even number. We also get the general formula term of the stability region of the coupling gain for different even multiples of time delay, and from this we easily obtain the shrink of the stability region as time delay increases. In addition, we explore the impact of slight perturbations in time delay on high frequency robustness.