Estimates for solutions of homogeneous time-delay systems: Comparison of Lyapunov-Krasovskii and Lyapunov-Razumikhin techniques

TL;DR

This paper compares Lyapunov-Krasovskii and Lyapunov-Razumikhin methods for estimating responses of nonlinear homogeneous time-delay systems, providing explicit constants and new bounds.

Contribution

It introduces new response estimates using Lyapunov-Krasovskii functionals and compares them with existing Razumikhin-based results, including explicit constants.

Findings

Lyapunov-Krasovskii approach yields new response estimates.

Explicit constants are derived for the estimates.

Comparison highlights strengths of both methods.

Abstract

In this contribution, the estimates for the response of time delay systems with nonlinear homogeneous right-hand side of degree strictly greater than one are constructed. The existing results obtained via the Lyapunov--Razumikhin approach are reminded. Their proofs, revisited in the appendix, lead to explicit expressions of the involved constants. Based on a recently introduced Lyapunov--Krasovskii functional and known estimates of the domain of attraction, we present new estimates of the system response. We compare both approaches and discuss the illustrative examples.