Dwell time for switched systems with multiple equilibria on a finite time-interval

TL;DR

This paper analyzes the behavior of solutions in switched systems with multiple stable equilibria, establishing conditions for stability and attractor properties based on switching frequency and dwell time.

Contribution

It introduces an ideal attractor concept and provides conditions under which solutions remain close to it, advancing understanding of stability in switched systems.

Findings

Solutions stay within an $oldsymbol{ ext{epsilon}}$-inflation of the ideal attractor if switchings are slow enough.

Conditions are given for the $oldsymbol{ ext{epsilon}}$-inflation to be a global attractor.

The effect of increased switchings on transition times between regions is analyzed.

Abstract

We describe the behavior of solutions of switched systems with multiple globally exponentially stable equilibria. We introduce an ideal attractor and show that the solutions of the switched system stay in any given $\varepsilon$-inflation of the ideal attractor if the frequency of switchings is slower than a suitable dwell time $T$. In addition, we give conditions to ensure that the $\varepsilon$-inflation is a global attractor. Finally, we investigate the effect of the increase of the number of switchings on the total time that the solutions need to go from one region to another.