A Perron–Frobenius Theorem for a Class of Positive Quasi-Polynomial Matrices

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Applied Mathematics Letters 19 (2006) 747–751 www.elsevier.com/locate/aml

APerron–Frobenius theorem for a class of positive quasi-polynomial matrices

Pham Huu Anh Ngoc∗

Department of Mathematics, The University of Electro-Communications, Chofu, Tokyo 182-8585, Japan

Received 26 July 2004; received in revised form 21 October 2005; accepted 27 October 2005

Abstract

In this work,wegiveanextension of the classical Perron–Frobenius theorem to positive quasi-polynomial matrices. Then the result obtained is applied to derive necessary and sufﬁcient conditions for the exponential stability of positive linear time-delay differentialsystems. c 2005 Elsevier Ltd. All rights reserved.

Keywords: Perron–Frobenius theorem; Positive system; Exponential stability of linear time-delay system; Quasi-polynomial matrices

1. Introduction

It is well known that the principal tool for the analysis of the stability and robust stability of a positive system is the Perron–Frobenius theorem; see, e.g. [1–3,5–9,11,12,15]. It is natural to ask whether it is possible to generalize this to classes of general (positive) linear systems and to positive nonlinear systems. To our knowledge, there are a large number of extensions of the classical Perron–Frobenius theorem in the literature. We were able to ﬁnd the latest extensions of the Perron–Frobenius theorem to real, complex matrices in [13,14,16]andtononlinear systems in [2, 10]. However, the investigation of extension of the classical Perron–Frobenius theorem to (positive) quasi-polynomial matrices associated with linear time-delay differential systems of the form n x˙(t) = A0x(t) + A1x(t − h1) + ···+ Am x(t − hm ), x(t) ∈ R , t ≥ 0, (1) has not been studied yet. In this work, we give an extension of the classical Perron–Frobenius theorem to positive quasi-polynomial matrices associated with positive linear time-delay differential systems of the form (1).Thenthe result obtained is applied to derive necessary and sufﬁcient conditions for exponential stability of positive systems of the form (1). The organization of this work is as follows. In the next section, we summarize some notation and recall the classical Perron–Frobenius theorem which will be used in the remainder. The main results of the work will be presented in Section 3 where we extend the classical Perron–Frobenius theorem to positive quasi-polynomial matrices.

∗ Tel.: +81 516 204613; fax: +81 516 227510. E-mail address: [email protected].

In Section 4,asanapplication of the Perron–Frobenius theorem of positive quasi-polynomial matrices, we give necessary and sufﬁcient conditions for the exponential stability of the positive linear time-delay differential systems of the form (1).

2. Preliminaries

Let K = C or R and n, l, q be positive integers. Inequalities between real matrices or vectors will be understood componentwise, i.e. for two real l × q-matrices A = (aij) and B = (bij),the inequality A ≥ B means aij ≥ bij for i = 1, ...,l, j = 1, ...,q.Furthermore,ifaij > bij for i = 1, ...,l, j = 1, ...,q then we write A B instead of l×q n A ≥ B.The set of all nonnegative l × q-matrices is denoted by R+ .Ifx = (x1, x2, ...,xn) ∈ K and P = (pij) ∈ l×q n×n K we deﬁne |x|=(|xi |) and |P|=(|pij|).Then|CD|≤|C||D|.Foranymatrix A ∈ K the spectral abscissa of the matrix A is denoted by µ(A) = max{Rλ : λ ∈ σ(A)},whereσ(A) := {z ∈ C : det(zIn − A) = 0} is the set n×n of all eigenvalues of A.LetA = (aij) ∈ R .Thematrix A is called a Metzler matrix if aij ≥ 0, for every i = j. It is equivalent to A + αIn being a nonnegative matrix for some α ∈ R.Thenext theorem summarizes some basic properties of Metzler matrices; see e.g. [7].

Theorem 2.1. Suppose that A ∈ Rn×n is a Metzler matrix. Then (i) (Perron–Frobenius theorem for Metzler matrices) µ(A) is an eigenvalue of A and there exists a nonnegative eigenvector x ≥ 0, x = 0 such that Ax = µ(A)x. (ii) Given α ∈ R,there exists a nonzero vector x ≥ 0 such that Ax ≥ αxifand only if µ(A) ≥ α. −1 (iii) (tIn − A) exists and is nonnegative if and only if t >µ(A). n×n n×n (iv) Given B ∈ R+ , C ∈ C ,then |C|≤B ⇒ µ(A + C) ≤ µ(A + B). (2)

3. A Perron–Frobenius theorem for a class of positive quasi-polynomial matrices

Consider a quasi-polynomial matrix associated with the linear time-delay differential system (1) of the form

−h1s −hm s Q(s) := sIn − A0 − A1e − ···− Ame , s ∈ C, (3) n×n where A0, A1, ...,Am ∈ R are given matrices and 0 < h1 < h2 < ···< hm are given positive numbers. For s ∈ C,ifdetQ(s) = 0thens is called an eigenvalue of the quasi-polynomial matrix (3).Then, a nonzero vector x ∈ Cn satisfying Q(s)x = 0iscalledaneigenvector of Q(·) corresponding to the eigenvalue s.Weset

µ0 := sup{Rs : det Q(s) = 0} (4) and µ0 is called the spectral abscissa of the quasi-polynomial matrix (3).Itiswell known that µ0 < +∞ and det Q(s) has at most ﬁnitely many zeros in a given vertical strip; see e.g. [4]. So, we can replace sup by max in (4).

Deﬁnition 3.1. The quasi-polynomial matrix (3) is called positive if A0 is a Metzler matrix and A1, ...,Am are nonnegative matrices. It is important to note that the quasi-polynomial matrix (3) is positive if and only if the associated linear delay-time n×n differentialsystem(1) is a positive system. That is, for any initial function φ ∈ C([−hm, 0], R+ ),the corresponding n×n solution x(t,φ),t ≥ 0ofthesystem (1) satisﬁes x(t,φ) ∈ R+ , ∀t ≥ 0. For further details, we refer the reader to [15]. We are now in the position to prove the main result of this work. The following theorem is an extension of Theorem 2.1 for the positive monomial matrix P(s) = sIn − A to positive quasi-polynomial matrices of the form (3).

Theorem 3.2. Letthe quasi-polynomial matrix (3) be positive. Then

(i) (Perron–Frobenius theorem for positive quasi-polynomial matrices) µ0 is an eigenvalue of the quasi-polynomial matrix (3) and there exists a nonnegative eigenvector x such that Q(µ0)x = 0. P. H.A. Ngoc / Applied Mathematics Letters 19 (2006) 747–751 749

n (ii) Given θ ∈ R,there exists a nonzero vector x ∈ R+ such that

−h θ −hm θ (A0 + A1e 1 + ···+ Ame )x ≥ θx

if and only if µ0 ≥ θ. (iii) For t ∈ R,wehave −1 t >µ0 ⇔ Q(t) ≥ 0. −h s Proof. (i) Assume that det Q(s) = 0, for some s ∈ C and Rs = µ0.Thisimplies that s ∈ σ(A0 + A1e 1 + ···+ −h s −h s Ame m ).Bythe definition of the spectral abscissa of a matrix, we derive that µ0 = Rs ≤ µ(A0 + A1e 1 + −h s ···+ Ame m ).ByTheorem 2.1(iv), it follows from A0 being a Metzler matrix and Ai , i ∈{1, 2, ...,m},being −h s −h s −h µ −h µ nonnegative matrices that µ(A0 + A1e 1 + ··· + Ame m ) ≤ µ(A0 + A1e 1 0 + ··· + Ame m 0 ).Thus, −h µ −h µ µ0 ≤ µ(A0 + A1e 1 0 + ···+ Ame m 0 ). −h t −h t Consider the continuous real function f (t) := t − µ(A0 + A1e 1 + ···+ Ame m ), t ∈ R.Then, we have f (µ0) ≤ 0. Assume f (µ0)<0. Since, clearly, limt→+∞ f (t) =+∞,wehave f (t0) = 0forsomet0 >µ0, −h t −h t so that t0 = µ(A0 + A1e 1 0 + ··· + Ame m 0 ).ByTheorem 2.1(i), t0 is an eigenvalue of the Metzler matrix −h t −h t A0 + A1e 1 0 + ···+Ame m 0 .Therefore, det Q(t0) = 0fort0 >µ0.However, this conflicts with the definition of −h µ −h µ µ0.Thus f (µ0) = 0, or equivalently µ0 = µ(A0 + A1e 1 0 + ···+ Ame m 0 ).Applying Theorem 2.1(i) to the −h µ −h µ Metzler matrix A0 + A1e 1 0 + ···+ Ame m 0 ,weget(i). −h θ −h θ n (ii) Assume that (A0 + A1e 1 + ···+ Ame m )x ≥ θx,forsome nonzero vector x ∈ R+.Then, it follows from −h θ −h θ Theorem 2.1(ii) that µ(A0 + A1e 1 + ···+ Ame m ) ≥ θ.Itisimportant to note that the function f given in the proof of (i) is strictly increasing on R.Since f (µ0) = 0and f (θ) ≤ 0, we derive µ0 ≥ θ.Conversely,ifµ0 ≥ θ −h θ −h θ then by Theorem 2.1(iv) µ(A0 + A1e 1 + ···+Ame m ) ≥ µ0 ≥ θ.Then, applying Theorem 2.1(ii) again to the −h θ −h θ Metzler matrix A0 + A1e 1 + ···+ Ame m ,wegetwhat we desired. −h µ −h µ (iii) For every t ∈ R, t >µ0,wehavet >µ0 = µ(A0 + A1e 1 0 + ···+ Ame m 0 ).Ontheother hand, from −h µ −h µ −h t −h t Theorem 2.1(iv), it follows that µ(A0 + A1e 1 0 + ···+Ame m 0 ) ≥ µ(A0 + A1e 1 + ···+Ame m ).Hence, −h t −h t −h t −h t t >µ(A0+ A1e 1 +···+Ame m ).Applying Theorem 2.1(iii) to the Metzler matrix A0+ A1e 1 +···+Ame m , we obtain Q(t)−1 ≥ 0. The converse also follows from Theorem 2.1(iii). This completes our proof.

Remark 3.3. From the argument of the proof of Theorem 3.2,weobserve that the continuous real function f (t) := −h t −h t t − µ(A0 + A1e 1 + ···+ Ame m ), t ∈ R is strictly increasing. So µ0 is the unique real solution of the equation f (t) = 0, t ∈ R.Thisfact is used in the next section.

4. Application to positive linear time-delay differential systems

We now apply the results obtained in the previous section to derive some necessary and sufficient conditions for the exponential stability of the positive linear time-delay differential systems of the form (1). n Recall that, for any initial function φ0 ∈ C := C([−hm , 0], R ),thesystem (1) has a unique solution x(·,φ0) defined and continuous on [−hm, ∞);see, e.g. [4]. Then, the system (1) is said to be exponentially stable if there are constants M > 0,α >0suchthat for all φ ∈ C, the solution x(·,φ)of (1) satisfies −α x(t,φ)≤Me t φ, t ≥ 0.

It is well known that the linear time-delay differential system (1) is exponentially stable if and only if µ0 := max{Rs : det Q(s) = 0} < 0; see, e.g. [4].

Theorem 4.1. Let the linear time-delay differential system (1) be positive. Then, the system (1) is exponentially stable if and only if the linear differential system without delay m x˙(t) = Ai x(t), t ≥ 0(5) i=0 is exponentially stable. 750 P. H.A. Ngoc / Applied Mathematics Letters 19 (2006) 747–751

Proof. The proof is immediate from Theorem 3.2 and Remark 3.3.

Theorem 4.2. Let the linear time-delay differential system (1) be positive. n (i) If the system (1) is exponentially stable then there exist a nonnegative vector p ∈ R+, p = 0 and a positive real number γ such that m T Ai p + γ p = 0. (6) i=0 (ii) Conversely, if the equality (6) holds for some strictly positive vector p ∈ Rn, p 0 and positive real number γ then the system (1) is exponentially stable. Proof. (i) Suppose that the positive system (1) is exponentially stable. By Theorem 4.1,thisisequivalent to the linear system without delay (5) being exponentially stable. From Theorem 2.1(i) (Perron–Frobenius), there exists a ∈ Rn, = m T = µ( m ) µ( m )< nonnegative vector p p 0, such that i=0 Ai p i=0 Ai p,where i=0 Ai 0. Hence, we getthe equality (6),whereγ =−µ( m A )>0. i=0 i ( m )T + γ = ∈ Rn, γ (ii) Let i=0 Ai p p 0forsomep p 0andpositive real number .Suppose to the contrary that µ( m ) ≥ the positive system (1) is not exponentially stable, or equivalently, i=0 Ai 0. Using again Theorem 2.1(i) ∈ Rn, = ( m ) = µ( m ) (Perron–Frobenius) there exists a nonnegative vector p0 p0 0suchthat i=0Ai p0 i=0 Ai p0. T( m )T = µ( m ) T T (µ( m ) + γ) = Thus p0 i=0 Ai i=0 Ai p0 .Taking(6) into account, we obtain p0 p i=0 Ai 0, or T = equivalently, p0 p 0. So we have reached a contradiction. This completes our proof.

Remark 4.3. (a) It is important to note that the nonnegative vector p ∈ Rn, p = 0satisfying (6) in (i) of Theorem 4.2 can be not strictly positive. To see this, we consider the following positive linear system without delay: x˙ (t) −10 x (t) 1 = 1 . (7) x˙2(t) 0 −2 x2(t) It is easy to check that though the positive linear system (7) is exponentially stable, there are not any p ∈ Rn, p 0, and positive real numbers γ satisfying the equality (6). Conversely,thevector p ∈ Rn in the converse statement (ii) of Theorem 4.2 must be strictly positive, p 0. To see this, we consider the following positive linear system: x˙ (t) 10 x (t) 1 = 1 . (8) x˙2(t) 0 −1 x2(t) Then, we have the equality (6) with p := (0, 1)T and γ = 1. But the system (8) is unstable. (b) The statement (ii) of Theorem 4.2 can be stated in the modiﬁed form: If there exist strictly positive vectors p, r ∈ Rn, p, r 0suchthat m T Ai p + r = 0 (9) i=0 then the system (1) is exponentially stable. Then we get back thesufﬁcient condition in Theorem 3.1 of [6]whereit wasproved by using a Lyapunov–Krasovskii functional.

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