Exponential stability of linear discrete systems with multiple delays by degenerated Lyapunov–Krasovskii functionals

Abstract

The problem of exponential stability of delayed discrete systems with multiple delays x(n+1)=(I+A)x(n)+∑i=1sBix(n−i),n=0,1,…is studied, where x=(x1x2…xm)T is an unknown vector, m and s are fixed positive integers, A, Bi are square constant matrices and I is a unit matrix. A new degenerated Lyapunov–Krasovskii functional is used to derive sufficient conditions for exponential stability and to derive an exponential estimate of the norm of solutions. Though often used in the study of stability, the assumption that the spectral radius of the matrix of linear terms is less than 1 is not applied here. The criterion derived is illustrated by an example and compared with previously known results.