A Perron-Frobenius Theorem for Positive Quasipolynomial Matrices Associated with Homogeneous Difference Equations

We extend the classical Perron-Frobenius theorem for positive quasipolynomial matrices associated with homogeneous difference equations. Finally, the result obtained is applied to derive necessary and sufficient conditions for the stability of positive system.


Introduction
Differential equations with dependence on the past state have various applications in physics, biology, economics, and so forth, and have also attracted many researchers as in the introduction and reference of [1][2][3][4].The neutral differential difference equation is one of general classes.And one of important neutral differential difference equations is where D(r,A) : C([−h;0],R n )→R n is linear continuous defined by here each A i is an n × n matrix, each r i is a constant satisfying r i > 0 and h = max {r i : i ∈ N}, N = {1, 2,...,N}, and y t : [−h;0]→R n is defined by y t (s) = y(t + s), s ∈ [−h;0].To get some properties of solution of (3.3), the stability of system (1.3) is required for the homogeneous difference equation of the form D(r,A)y t = 0, t ≥ 0; see [2].This equation can be rewritten as follows: We recall the definition in [2]: "the operator D(r,A) or the system (1.3) is called stable if the zero solution of (1.3) is uniformly asymptotically stable."Associated with the system (1.3), we define the quasipolynomial For s ∈ C, if det H(s) = 0, then s is called a characteristic root of the quasipolynomial matrix (1.4).Then, a nonzero vector , the spectral set of (1.4), and a H = sup{Rλ : λ ∈ σ(H(•))}, the spectral abscissa of (1.4) (noticing that we cannot replace sup by max).The following theorem is a wellknown result in [2].
Theorem 1.1.The system (1.3) is stable if and only if a H < 0.
Thus, studying the stability of the system (1.3) turns out considering the characteristics of quasipolynomial (1.4).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 [5][6][7][8].In this work, we give an extension of the classical Perron-Frobenius theorem to positive quasipolynomial matrices of the form (1.4).Then the result obtained is applied to derive necessary and sufficient conditions for stability of positive systems of the form (1.3).An outline of this paper is as follows.In the next section, we summarize some notations and recall the classical Perron-Frobenius theorem which will be used in the remainder.The main results will be addressed in Section 3, where we extend the classical Perron-Frobenius theorem to positive quasipolynomial matrices.Finally, we apply the obtained results to give necessary and sufficient conditions for the stability of the positive systems of the form (1.3).

Preliminaries
We first introduce some notations.Let n, l, q be positive integers, a matrix P = [p i j ] ∈ R l×q is said to be nonnegative (P ≥ 0) if all its entries p i j are nonnegative.It is said to be positive (P > 0) if all its entries p i j are positive.For P,Q ∈ R l×q , P > Q means that P − Q > 0. The set of all nonnegative l × q-matrices is denoted by R l×q .A similar notation will be used for vectors.Let K = C or R, then for any x ∈ K n and P ∈ K l×q , we defined For any matrix A ∈ K n×n the spectral radius and the spectral abscissa of A are defined by r(A) = max {|λ| : λ ∈ σ(A)} and μ(A) = max {Rλ : λ ∈ σ(A)}, respectively, where σ(A) is the spectrum of A. We recall some useful results; see [9].B. T. Anh and D. D. X. Thanh 3 Theorem 2.1 (Perron-Frobenius).Suppose that A ∈ R n×n + .Then, (i) r(A) is an eigenvalue of A and there is a nonnegative eigenvector x ≥ 0, x =0 such that Ax = r(A)x; (ii) if λ ∈ σ(A) and |λ| = r(A), then the algebraic multiplicity of λ is not greater than the algebraic multiplicity of the eigenvalue r(A); (iii) given α > 0, there exists a nonzero vector x ≥ 0 such that Ax ≥ αx if and only if r(A) ≥ α; (iv) (tI − A) −1 exists and is nonnegative if and only if t > r(A). (2.1)

Main results
In this section, we will extend to Perron-Frobenius for positive quasipolynomial matrices of the form (1.4).
Proof.Assume that (λ m ) m∈N is a sequence in σ(H(•)) such that limRλ m = a H .Then, for every m ∈ N, there exists x m ∈ C n such that It is clear that f (•) is continuous and strictly decreasing on R and lim t→+∞ r(t) = 0.Moreover, from (3.3) we have f (Rλ m ) ≥ 1.Thus, from the continuity of f (•), there exists α m ≥ Rλ m satisfying r( N k=1 e −αmrk A k ) = 1 which implies α m ∈ σ(H(•)).Briefly, we have constructed a real sequence (α m ) that has the following properties: α m ∈ σ(H(•)) and α m ≥ Rλ m , for all m ∈ N.This follows from limα m = a H . Furthermore, σ(H(•)) is closed.Thus, a H ∈ σ(H(•)).The proof is complete.Theorem 3.2.Let quasipolynomial (1.4) be positive.Then, (i) a H is a characteristic root of H(a H ) and there is a nonnegative eigenvector x ≥ 0, x =0 such that H(a H )x = 0; (ii) given α > 0, there exists a nonzero vector x ≥ 0 such that ( N k=1 e −αrk A k )x ≥ x if and only if a H ≥ α; (iii) for t ∈ R, then one has (3.5) Proof.(i) First, we recall that in the proof of Theorem 3.1, we have constructed a real sequence (α n ) that has the following properties: α n ∈ σ(H(•)), for all n ∈ N, limα n = a H , and r( N k=1 e −αnrk A k ) = 1.Thus, this implies r( N k=1 e −aH rk A k ) = 1.By Theorem 2.1, there exists a nonnegative eigenvector x ≥ 0, x =0 such that ( N k=1 e −aH rk A k )x = x, or [I − ( N k=1 e −aH rk A k )]x = 0, then we get (i).
(ii) Assume that there exists a nonzero vector Therefore, applying Theorem 2.1 for positive matrix N k=1 e −αrk A k , we get (ii).(iii) For t > a H , and the decrease of Remark 3.3.From Theorem 3.2, it is easy to see that under the positivity assumption, the spectral abscissa, a H , is continuous with respect to the delay parameters.This is not the case if the positivity assumption is dropped; see [1,3].

Application to positive homogeneous difference equation
We now apply the results obtained in the previous section to derive some necessary and sufficient conditions for the stability of the positive homogeneous difference equation of the form (1.3).The following results are obtained by applying directly Theorems 3.1 and 3.2.
Theorem 4.1.Let quasipolynomial (1.4)  Proof.Assume that (1.3) is stable and positive.By (iii) in Theorem 3.2, from a H < 0, we obtain (ii) It is clear that in the positive system, applying Theorem 4.2 to check the existing stability for delay-difference equations is much more simpler and easier than the general results in [2][3][4].
Next, we address some remarks about the relationship between independent-delay stability and stability of the system (1.3).Definition 4.4.The system (1.3) is stable globally in the delays if it is stable for each The concept of global stability, which is sometimes called independent-delay stability, has interested many researchers as in [2,4,10] and references therein.In fact, if the system is globally stable, then it is stable, but the conversion is not true.However, in the case of positive system, both concepts are the same.This is stated in implications following corollary of Theorem 4.2.
Corollary 4.5.Let A i be all nonnegative matrices for all i ∈ N.Then, the following statements are equivalent: (i) the system (1.3)

3 )
Now let us define the function f : R→R by setting in Theorem 3.2, a H < 0. This completes the proof.B. T. Anh and D. D. X. Thanh 5 Remark 4.3.(i) The stability of positive systems do not depend on delay parameters.
be positive.The homogeneous difference equation (1.3) is stable if and only if all characteristic roots of H(•) lie inside a half plane C− = {λ ∈ C : Rλ < 0}.