Stability criteria of linear neutral systems with a single delay

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Abstract

This article studies the asymptotic stability of linear neutral delay-differential systems. Using the characteristic equation of the system, new simple delay-independent stability criteria are derived in terms of the spectral radius of modulus matrices. Numerical examples are given to demonstrate the validity of our main criteria.

Introduction

During the past years, the problem of stability analysis for neutral delay-differential systems has received great interest in the literature. A number of stability criteria based on the characteristic equation approach, involving the determination of eigenvalues, measures and norms of matrices, or matrix conditions in terms of Hurwitz matrices, have been presented by Stroinski [1], Hale et al. [2], Li [3], Hale and Verduyn Lunel [4], Hu and Hu [5], Bellen et al. [6] and Park and Won [7]. Some stability criteria (delay-independent and/or delay-dependent) are given in terms of the Lyapunov function and matrix inequalities (see, for example, Khusainov and Yun’kova [8], Lien et al. [9], Agarwal and Grace [10], Fridman [11], Lien [12] and Niculescu [13]). Based on the linear matrix inequality (LMI) approach, stability conditions have been developed by Park and Won [14] and Park [15] to make the criteria less conservative. Recently, to complement the stability criteria derived by Stroinski [1], Li [3], Hu and Hu [5] and Bellen et al. [6], the algebraic stability conditions via Routh–Hurwitz and Schur–Cohn criteria have been derived by Hu et al. [16].

This article deals with the asymptotic stability of linear neutral systems with a single time delay. Scalar inequalities involving the spectral radius and modulus matrices constitute the mathematical foundations of our approach. Using the characteristic equation of the system, new delay-independent stability criteria are derived. Numerical examples are given to demonstrate the validity of our main criteria and to compare them with the existing ones.

Section snippets

Notations and preliminaries

Let Rn(Cn) denote the n-dimensional real (complex) space and Rn×n (Cn×n) denote the set of all real (complex) n by n matrices. I denotes the unit matrix of appropriate order. λj(A) and ρ(A) denote the jth eigenvalue and the spectral radius of A, respectively. |A| denote the modulus matrix of A; AB represents that the elements of A and B, satisfy the inequality aijbij for all i and j. ∥A∥(:=λmax(A*A)) and μ(A)(:=12λmax(A+A*)) denote the spectral norm and the matrix measure of A, respectively.

Main results

Define F(s)=(sIA)−1 and let Fm denote the matrix formed by taking the maximum magnitude of each element of F(s) for R(s)⩾0. Since the system matrix A is Hurwitz, the matrix Fm exists and can be obtained for some s on imaginary axis by maximum modulus theorem. Furthermore, defineW=CA+B,H0=|W|+(I−|C|)−1|CW|,R=AC+B,K0=|R|+|RC|(I−|C|)−1.

Lemma 3.1

The neutral system (1) is asymptotically stable if A is a Hurwitz matrix, ρ(|C|)<1 andsupρ[F(s)(I−ξ(s)C)−1Wξ(s)]<1∀s∈CsuchthatR(s)⩾0,where F(s)=(sIA)−1, ξ(s)=exp(−

Illustrative examples

We will compare our new criteria with the following criteria for asymptotic stability of system (1) in the case of |C|<1.Criterion1[5]:k1≜μ(A)+∥B∥+∥CA∥+∥CB∥1−∥C∥<0.Criterion2[3]:k2≜μ(A)+∥B∥+∥C∥∥A∥+∥C∥∥B∥1−∥C∥<0.Criterion3[6]:k3≜μ(A)+∥AC+B∥1−∥C∥<0.Criterion4(Theorem3.3in[16]):k4≜ρ(G0)<1andρ(|N|)<1.Criterion5(Theorem3.4in[16]):k5≜ρ(Gq)<1forsomeintegerq>1andρ(|N|)<1,whereG0=|L|+|M|+(I−|N|)−1(|NL|+|NM|),Gq=|L|+|M|+∑j=1q(|NjL|+|NjM|)+(I−|N|)−1(|Nq+1L|+|Nq+1M|)andL=(I−A)−1(B+C),M=(I−A)−1(B−C),N=(I−A)

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