Estimates for weighted homogeneous delay systems: A Lyapunov-Krasovskii-Razumikhin approach

TL;DR

This paper introduces a new method combining Lyapunov-Krasovskii and Razumikhin techniques to estimate solutions and attraction domains of weighted homogeneous delay systems, resulting in less conservative bounds.

Contribution

It generalizes existing Lyapunov-Krasovskii functionals for weighted homogeneous systems and develops a novel approach integrating Razumikhin ideas for improved estimates.

Findings

The new approach provides less conservative estimates than classical methods.

A lower bound for the functional is constructed on a Razumikhin-inspired set.

An example demonstrates the effectiveness of the proposed method.

Abstract

In this paper, we present estimates for solutions and for the attraction domain of the trivial solution for systems with delayed and nonlinear weighted homogeneous right-hand side of positive degree. The results are achieved via a generalization of the Lyapunov-Krasovskii functional construction presented recently for homogeneous systems with standard dilation. Along with the classical approach for the calculation of the estimates within the Lyapunov-Krasovskii framework, we develop a novel approach which combines the use of Lyapunov-Krasovskii functionals with ideas of the Razumikhin framework. More precisely, a lower bound for the functional on a special set of functions inspired by the Razumikhin condition is constructed, and an additional condition imposed on the solution of the comparison equation ensures that this bound can be used to estimate all solutions in a certain neighbourhood of the trivial one. An example shows that this approach yields less conservative estimates in comparison with the classical one.