Exponential Stability of Linear Systems with Multiple Time Delays

Abstract

In this paper, a class of linear systems with multiple time delays is studied. The problem of exponential stability of time-delay systems has been investigated by using Lyapunov functional method. We will convert the system of multiple time delays into a single time delay system and show that if the old system is stable then the new one is so. Then we investigate the stability of converted new system, by using matrix decomposition and linear matrix inequality (LMI) technique. Some numerical examples are given to illustrate the efficiency of our method.

Impulse Delay in the Cardiac Conduction System

“…where I is the identity matrix. The solution of (37) with ( 38) is of the form [28] 0 0 It was shown in [28] that the fundamental solution matrix of equation (37) with identity initial conditions (38) has the form…”

“…The goal here is to convert the system with multiple time delays to a stable system with single time-delay. When this is done, the multiple time delay case may be handled as a single delay system, as was done in Ordokhani et al [37].…”

“…Lemma 2 [38]The system (70) is said to be exponentially stable with decay rate α, if there is a function𝜁𝜁: ℝ n → ℝ n such that for each 𝜑𝜑(t) ∈C( [-τ, 0], ℝ n ) the solution x(t,𝜑𝜑) of the system satisfies ||x(t,𝜑𝜑)||≤ 𝜁𝜁(||A||)e -αt , ∀t ∈ℝ + . Lemma3 [37] System (70) is uniformly asymptotically stable independent of delay if…”