Positive Forms and Stability of Linear Time-Delay Systems

TL;DR

This paper develops a method to analyze the stability of linear systems with delays by constructing positive Lyapunov functions through semidefinite programming, enabling efficient stability verification.

Contribution

It provides an explicit parametrization of positive Lyapunov functions for delay systems, facilitating stability analysis via semidefinite programming.

Findings

Stability analysis formulated as a semidefinite program

Explicit parametrization of positive Lyapunov functions

Applicable to linear time-delay systems

Abstract

We consider the problem of constructing Lyapunov functions for linear differential equations with delays. For such systems it is known that exponential stability implies the existence of a positive Lyapunov function which is quadratic on the space of continuous functions. We give an explicit parametrization of a sequence of finite-dimensional subsets of the cone of positive Lyapunov functions using positive semidefinite matrices. This allows stability analysis of linear time-delay systems to be formulated as a semidefinite program.