On Switching Stabilizability for Continuous-Time Switched Linear Systems

TL;DR

This paper establishes the equivalence of four types of switching stabilizability for continuous-time switched linear systems, characterized by a piecewise quadratic control-Lyapunov function, advancing the theoretical understanding of stabilization conditions.

Contribution

It proves the equivalence of different stabilizability definitions and characterizes them via a specific piecewise quadratic Lyapunov function, providing a unified theoretical framework.

Findings

All four stabilizability definitions are equivalent.

Existence of a piecewise quadratic control-Lyapunov function is necessary and sufficient.

The Lyapunov function can be expressed as the minimum of finitely many quadratic functions.

Abstract

This paper studies switching stabilization problems for continuous-time switched linear systems. We consider four types of switching stabilizability defined under different assumptions on the switching control input. The most general switching stabilizability is defined as the existence of a measurable switching signal under which the resulting time-varying system is asymptotically stable. Discrete switching stabilizability is defined similarly but requires the switching signal to be piecewise constant on intervals of uniform length. In addition, we define feedback stabilizability in Filippov sense (resp. sample-and-hold sense) as the existence of a feedback law under which closed-loop Filippov solution (resp. sample-and-hold solution) is asymptotically stable. It is proved that the four switching stabilizability definitions are equivalent and their sufficient and necessary condition is the existence of a piecewise quadratic control-Lyapunov function that can be expressed as the pointwise minimum of a finite number of quadratic functions.