A Lyapunov functional for a system with both lumped and distributed delay

Abstract

In the paper construction of a Lyapunov functional for time delay system with both lumped and distributed delay is presented. The Lyapunov functional is determined by means of the Lyapunov matrix. The method of determination of the Lyapunov matrix for time delay system with both lumped and distributed delay is presented. It is given the example illustrating the method.

A Lyapunov functional for a neutral system with a distributed time delay

“…Presented method allows achieving the analytical formulas on the Lyapunov functional coefficients. This article extends the results presented in [6] to a neutral system with both lumped and distributed time delay. To find the Lyapunov functional coefficients we need to solve a set of differential equations (4.29).…”

“…In the paper [4] was introduced a Lyapunov functional determination method for a linear system with two delays both retarded and neutral type and in [5] for a linear neutral system with k-non-commensurate time delays. In the paper [6] was proposed a Lyapunov quadratic functional determination method for a linear system with both lumped and distributed time delay and in [7] was stated such method for a system with a time-varying delay. Infante and Castelan's [10] construction of a Lyapunov functional is based on a solution of a matrix differential-difference equation on a finite time interval.…”

“…The presented method gives analytical formulas for the Lyapunov functional coefficients. The novelty of the paper lies in the extension of the results presented in [6] to a neutral system with both lumped and distributed time delay. To the best of author's knowledge, such extension has not been reported in the literature.…”

Lyapunov matrices approach to the parametric optimization of a neutral system

“…for t t 0 , where x(t, φ) is a solution of system (1), with the initial function φ ∈ PC([−r, 0], R n ), given by (9) and x t (φ) is a shifted restriction of x(·, φ) to an interval [t − r,t], given by (5).…”

“…The method of determination of a Lyapunov functional for a time delay system with one delay was presented by Repin [15]. Duda [3] used the Repin's method to determination of a Lyapunov functional for a system with two delays, for a system with both lumped and distributed delay [5,10], for a system with time-varying delay [6,8]. In last years a method of determination of a Lyapunov functional by means of Lyapunov matrices is very popular, see for example [9,12,13,14,16].…”