The Robustness of Strong Stability of Positive Homogeneous Difference Equations

Abstract

We study the robustness of strong stability of the homogeneous difference equation via the concept of strong stability radii: complex, real and positive radii in this paper. We also show that in the case of positive systems, these radii coincide. Finally, a simple example is given.

1. Introduction

Motivated by many applications in control engineering, problems of robust stability of dynamical systems have attracted a lot of attention of researchers during the last twenty years. In the study of these problems, the notion of stability radius was proved to be an effective tool, see [1–5]. In this paper, we study the robustness of strong stability of the homogeneous difference equation under parameter perturbations.

The organization of this paper is as follows. In Section 2, we recall some results on nonnegative matrices and present preliminary results on homogeneous equations for later use. In Section 3, we study a complex strong stability radius under multiperturbations. Next, we present some results on strong stability radii of the positive class equations under parameter perturbations. It is shown that complex, real, and positive strong stability radii of positive systems coincide. More important, estimates and computable formulas of these stability radii are also derived. Finally, a simple example is given.

2. Preliminaries

2.1. Nonnegative Matrices

We first introduce some notations. Let 𝑛,𝑙,𝑞 be positive integers, a matrix 𝑃=[𝑝_𝑖𝑗]∈ℝ^𝑙×𝑞 is said to be nonnegative (𝑃≥0) if all its entries 𝑝_𝑖𝑗 are nonnegative. It is said to be positive (𝑃>0) if all its entries 𝑝_𝑖𝑗 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 define |𝑥|∈ℝ^𝑛₊ and |𝑃|∈ℝ₊^𝑙×𝑞 by |𝑥|=(|𝑥_𝑖|),|𝑃|=[|𝑝_𝑖𝑗|]. For any matrix 𝐴∈𝕂^𝑛×𝑛 the spectral radius and the spectral abscissa of 𝐴 is 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 [6].

A norm ‖⋅‖ on 𝕂^𝑛 is said to be monotonic if it satisfies|𝑥|≤|𝑦|⇒‖𝑥‖≤‖𝑦‖,∀𝑥,𝑦∈𝕂^𝑛. (2.1)It can be shown that a vector norm ‖⋅‖ on 𝕂^𝑛 is monotonic if and only if ‖𝑥‖=‖|𝑥|‖ for all 𝑥∈𝕂^𝑛, see [7]. All norms on 𝕂^𝑛 we use in this paper are assumed to be monotonic. 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).Theorem 2.2. Let 𝐴∈𝕂^𝑛×𝑛,𝐵∈ℝ₊^𝑛×𝑛. If |𝐴|≤𝐵 then𝑟(𝐴)≤𝑟(|𝐴|)≤𝑟(𝐵). (2.2)

2.2. Homogeneous Difference Equations

Consider the neutral differential difference equation of the following form:𝑑𝑑𝑡[𝐷(𝑟,𝐴)𝑦_𝑡]=𝑓(𝑡,𝑦_𝑡), (2.3)where 𝐷(𝑟,𝐴)∶𝐶([−ℎ;0],ℝ^𝑛)→ℝ^𝑛 is linear continuous defined by𝐷(𝑟,𝐴)𝜙=𝜙(0)−^𝑁_𝑖=1𝐴_𝑖𝜙(−𝑟_𝑖),𝜙∈𝐶([−ℎ;0],ℝ^𝑛). (2.4)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∈𝐶([−ℎ;0],ℝ^𝑛) is defined by 𝑦_𝑡(𝑠)=𝑦(𝑡+𝑠),𝑠∈[−ℎ;0],𝑡≥0. Recall that there is a strictly close relation between the asymptotic behavior of solutions of (2.3) and that of associated linear homogeneous difference equations𝐷(𝑟,𝐴)𝑦_𝑡=0,𝑡≥0, (2.5)or equivalently,𝑦(𝑡)−^𝑁_𝑖=1𝐴_𝑖𝑦(𝑡−𝑟_𝑖)=0. (2.6)A study of the asymptotic behavior of solutions of system (2.6) plays a fundamental role in understanding the asymptotic behavior of solutions of linear neutral differential equations of the form (2.3), see [8].

We recall the definition in [8]: the operator 𝐷(𝑟,𝐴) or system (2.6) is called stable if the zero solution of (2.6) with 𝑦₀∈𝐶_𝐷(𝑟,𝐴)={𝜙∈𝐶([−ℎ,0],ℝ^𝑛)∶𝐷(𝑟,𝐴)𝜙=0} is uniformly asymptotically stable.

Associated with system (2.6) we define the quasipolynomial𝐻(𝑠)=𝐼−^𝑁_𝑘=1𝑒^{−𝑠𝑟𝑘}𝐴_𝑘. (2.7)For 𝑠∈ℂ, if det𝐻(𝑠)=0, then 𝑠 is called a characteristic root of the quasipolynomial matrix (2.7). Then, a nonzero vector 𝑥∈ℂ^𝑛 satisfying 𝐻(𝑠)𝑥=0 is called an eigenvector of 𝐻(⋅) corresponding to the characteristic root 𝑠. We set 𝜎(𝐻(⋅))={𝜆∈ℂ∶det𝐻(𝜆)=0}, the spectral set of (2.7), and 𝑎_𝐻=sup{ℜ𝜆∶𝜆∈𝜎(𝐻(⋅))}, the spectral abscissa of (2.7). The following lemma is a well-known result in [8]. Theorem 2.3. System (2.6) is stable if and only if 𝑎_𝐻<0.It is well known that 𝑎_𝐻 is not continuous in the delays (𝑟₁,…,𝑟_𝑁), see [9]. One consequence of the noncontinuity is that arbitrarily small perturbations on the delays may destroy stability of the difference equation. This has led to the introduction of the concept of strong stability in Hale and Verduyn Lunel [10]. Definition 2.4. System (2.6) is strongly stable in the delays if it is stable for each (𝑟_𝑖)_𝑖∈𝑁∈ℝ^𝑁₊.The concept of strong stability has interested many researchers as in [8–13] and references therein. Now we recall a result in [10]. Theorem 2.5. The following statements are equivalent:
(i)system (2.6) is strongly stable,(ii)sup{𝑟(∑^𝑁_𝑖=1𝑧_𝑖𝐴_𝑖)∶|𝑧_𝑖|=1,𝑖∈𝑁}<1.We set ℂ₁={𝑧∈ℂ∶|𝑧|<1} and 𝜕ℂ₁={𝑧∈ℂ∶|𝑧|=1}. Since 𝑟(⋅) is continuous in ℂ^𝑛×𝑛, we imply the continuity of the following function 𝑔∶(𝜕ℂ₁)^𝑁→ℝ defined by𝑔(𝑧₁,…,𝑧_𝑛)=𝑟(^𝑁_𝑖=1𝑧_𝑖𝐴_𝑖). (2.8)Moreover, by the compactness of the set (𝜕ℂ₁)^𝑁, there exists 𝑧^∗=(𝑧^∗₁,…,𝑧^∗_𝑁) such that𝑟(^𝑁_𝑖=1𝑧^∗_𝑖𝐴_𝑖)=sup{𝑟(^𝑁_𝑖=1𝑧_𝑖𝐴_𝑖)∶|𝑧_𝑖|=1,𝑖∈𝑁}. (2.9)By the above result, we can get the following statement: system (2.6) is strongly stable if and only ifmax{𝑟(^𝑁_𝑖=1𝑧_𝑖𝐴_𝑖)∶(𝑧_𝑖)_𝑖∈𝑁∈(𝜕ℂ₁)^𝑁}<1. (2.10)

3. Main Results

3.1. Complex Strong Stability Radius

Suppose that system (2.6) is strongly stable. Now we assume that each matrix 𝐴_𝑖 is subjected to the perturbation of the form𝐴_𝑖→𝐴_𝑖+𝐷_𝑖Δ_𝑖𝐸_𝑖,𝑖∈𝑁, (3.1)where 𝐷_𝑖∈ℂ^{𝑛×𝑝𝑖},𝐸_𝑖∈ℂ^{𝑞𝑖×𝑛} are given matrices defining the structure of the perturbations and Δ_𝑖∈ℂ^{𝑝𝑖×𝑞𝑖} are unknown matrices, 𝑖∈𝑁. We write the perturbed system𝑦(𝑡)−^𝑁_𝑖=1(𝐴_𝑖+𝐷_𝑖Δ_𝑖𝐸_𝑖)𝑦(𝑡−𝑟_𝑖)=0. (3.2)Definition 3.1. Let system (2.6) be strongly stable. The complex, real, and positive strong stability radii of system (2.6) under perturbations of the form (3.1) are defined by𝑟_ℂ=inf{^𝑁_𝑖=1‖Δ_𝑖‖∶Δ_𝑖∈ℂ^{𝑝𝑖×𝑞𝑖},𝑖∈𝑟𝑁,system(3.2)isnotstronglystable},_ℝ=inf{^𝑁_𝑖=1‖Δ_𝑖‖∶Δ_𝑖∈ℝ^{𝑝𝑖×𝑞𝑖},𝑖∈𝑟𝑁,system(3.2)isnotstronglystable},₊=inf{^𝑁_𝑖=1‖Δ_𝑖‖∶Δ_𝑖∈ℝ^{𝑝𝑖×𝑞𝑖}₊,𝑖∈𝑁,system(3.2)isnotstronglystable}, (3.3)respectively, we set inf∅=+∞.If system (2.6) is strongly stable, we define a function 𝐻(⋅,⋅)∶ℂ⧵ℂ₁×(𝜕ℂ₁)^𝑁→ℂ^𝑛×𝑛 by 𝐻(𝜆,𝑧)=(𝜆𝐼−∑^𝑁_𝑖=1𝑧_𝑖𝐴_𝑖)⁻¹,𝜆∈ℂ⧵ℂ₁,𝑧=(𝑧_𝑖)_𝑖∈𝑁∈(𝜕ℂ₁)^𝑁. It is easy to see that 𝐻(⋅,⋅) is well-defined. For any 𝜆∈ℂ⧵ℂ₁,𝑧=(𝑧_𝑖)_𝑖∈𝑁∈(𝜕ℂ₁)^𝑁, we set𝐺_𝑖𝑗(𝜆,𝑧)=𝐸_𝑖𝐻(𝜆,𝑧)𝐷_𝑗∈ℂ^{𝑞𝑖×𝑝𝑗}. (3.4)Theorem 3.2. Let system (2.6) be strongly stable. Then we have
(i)1max_{𝑖,𝑗∈𝑁}sup_{|𝜆|≥1,𝑧∈(𝜕ℂ1)𝑁}‖𝐺_𝑖𝑗(𝜆,𝑧)‖≤𝑟_ℂ≤1max_𝑖∈𝑁sup_{|𝜆|≥1,𝑧∈(𝜕ℂ1)𝑁}‖𝐺_𝑖𝑖(𝜆,𝑧)‖, (3.5)(ii)in particular, if 𝐷_𝑖=𝐷_𝑗 (or 𝐸_𝑖=𝐸_𝑗) for all 𝑖,𝑗∈𝑁, then we have𝑟_ℂ=1max_𝑖∈𝑁sup_{|𝜆|≥1,𝑧∈(𝜕ℂ1)𝑁}‖𝐺_𝑖𝑖(𝜆,𝑧)‖. (3.6)Proof. Let Δ=(Δ_𝑖)_𝑖∈𝑁 be a destabilizing disturbance. Then there exists (𝜆,𝑧)∈ℂ⧵ℂ₁×(𝜕ℂ₁)^𝑁 such that 𝜆∈𝜎(∑^𝑁_𝑖=1𝑧_𝑖(𝐴_𝑖+𝐷_𝑖Δ_𝑖𝐸_𝑖)). This means that there exists a nonzero vector 𝑥 satisfying(^𝑁_𝑖=1𝑧_𝑖(𝐴_𝑖+𝐷_𝑖Δ_𝑖𝐸_𝑖))𝑥=𝜆𝑥. (3.7)This follows that(^𝑁_𝑖=1𝑧_𝑖𝐷_𝑖Δ_𝑖𝐸_𝑖)𝑥=(𝜆𝐼−^𝑁_𝑖=1𝑧_𝑖𝐴_𝑖)𝑥, (3.8)or equivalently,𝑥=^𝑁_𝑖=1𝑧_𝑖𝐻(𝜆,𝑧)𝐷_𝑖Δ_𝑖𝐸_𝑖𝑥. (3.9)Choose 𝑞∈𝑁 such that ‖𝐸_𝑞𝑥‖=max{‖𝐸_𝑖𝑥‖∶𝑖∈𝑁}. Multiplying the above equation with 𝐸_𝑞, we obtain𝐸_𝑞𝑥=^𝑁_𝑖=1𝑧_𝑖𝐸_𝑞𝐻(𝜆,𝑧)𝐷_𝑖Δ_𝑖𝐸_𝑖𝑥=^𝑁_𝑖=1𝑧_𝑖𝐺_𝑞𝑖(𝜆,𝑧)Δ_𝑖𝐸_𝑖𝑥. (3.10)This implies that‖𝐸_𝑞𝑥‖≤^𝑁_𝑖=1‖𝑧_𝑖‖‖𝐺_𝑞𝑖(𝜆,𝑧)‖‖Δ_𝑖‖‖𝐸_𝑖≤𝑥‖_𝑁𝑖=1max_{𝑖,𝑗∈𝑁}sup_{|𝜆|≥1,𝑧∈(𝜕ℂ1)𝑁}‖𝐺_𝑖𝑗(𝜆,𝑧)‖‖Δ_𝑖‖‖𝐸_𝑞𝑥‖. (3.11)From this inequality and the definition of 𝑟_ℂ, the left-hand inequality of (i) follows:𝑟_ℂ≥1max_{𝑖,𝑗∈𝑁}sup_{|𝜆|≥1,𝑧∈(𝜕ℂ1)𝑁}‖𝐺_𝑖𝑗(𝜆,𝑧)‖. (3.12)Now it remains to prove the right-hand inequality of (i): 𝑟_ℂ≤1max_𝑖∈𝑁sup_{|𝜆|≥1,𝑧∈(𝜕ℂ1)𝑁}‖𝐺_𝑖𝑖(𝜆,𝑧)‖. (3.13)Indeed, for any (𝜆,𝑧)∈ℂ⧵ℂ₁×(𝜕ℂ₁)^𝑁, and 𝑖∈𝑁, there exists nonzero vector 𝑥∈ℂ^𝑝𝑖 such that ‖𝑥‖=1 and ‖𝐺_𝑖𝑖(𝜆,𝑧)𝑥‖=‖𝐺_𝑖𝑖(𝜆,𝑧)‖. By Hahn-Banach theorem, there exists 𝑦^∗∈(ℂ^𝑞𝑖)^∗ satisfying ‖𝑦^∗‖=1 and 𝑦^∗(𝐺_𝑖𝑖(𝜆,𝑧)𝑥)=‖𝐺_𝑖𝑖(𝜆,𝑧)𝑥‖. We define a matrix Δ∈ℂ^{𝑝𝑖×𝑞𝑖} by setting1Δ=‖𝐺_𝑖𝑖(𝜆,𝑧)‖𝑥𝑦^∗∈ℂ^{𝑝𝑖×𝑞𝑖}. (3.14)Now we construct the disturbance Δ=(Δ₁,…,Δ_𝑁) defined by Δ_𝑗=𝑧_𝑖⁻¹Δ,𝑗=𝑖,0,𝑗≠𝑖. (3.15) It is easy to check that ∑^𝑁_𝑖=1‖Δ_𝑖‖=‖Δ‖=1/‖𝐺_𝑖𝑖(𝜆,𝑧)‖. Moreover, we have𝜆𝑥=(^𝑁_𝑖=1𝑧_𝑖(𝐴_𝑖+𝐷_𝑖Δ_𝑖𝐸_𝑖))𝑥, (3.16)where 𝑥=𝐻(𝜆,𝑧)𝐷_𝑖𝑥. This means that Δ is a destabilizing disturbance. Thus,𝑟_ℂ≤1max_𝑖∈𝑁sup_{|𝜆|≥1,𝑧∈(𝜕ℂ1)𝑁}‖𝐺_𝑖𝑖(𝜆,𝑧)‖. (3.17)The proof of (i) is complete, and (ii) can be obtained directly from (i).

In general, the complex, real, and positive radius are distinct, see [4, 5]. Theorem 3.2 reduces the computation of the complex strong stability radius to a global optimization problem with many variations while the problem for the real stability radius is much more difficult, see [5]. It is therefore natural to investigate for which kind of systems these three radii coincide. The answer will be found in the next section.

3.2. Strong Stability Radii of Positive Systems

In this section, we restrict system (2.6) to be positive, that is, 𝐴_𝑖 are nonnegative for all 𝑖∈𝑁.Lemma 3.3. Let 𝐴_𝑖∈ℝ₊^𝑛×𝑛. Then we have
(i)𝑟(∑^𝑁_𝑖=1𝐴_𝑖)=sup_{𝑧∈(𝜕ℂ1)}^𝑁𝑟(∑^𝑁_𝑖=1𝑧_𝑖𝐴_𝑖);(ii)𝑟(∑^𝑁_𝑖=1𝐴_𝑖)<𝑡₁≤𝑡₂⇒0≤(𝑡₂𝐼−∑^𝑁_𝑖=1𝐴_𝑖)⁻¹≤(𝑡₁𝐼−∑^𝑁_𝑖=1𝐴_𝑖)⁻¹,Proof. (i) By Theorem 2.2, we have𝑟(^𝑁_𝑖=1𝑧_𝑖𝐴_𝑖)≤𝑟(^𝑁_𝑖=1|𝑧_𝑖𝐴_𝑖|)=𝑟(^𝑁_𝑖=1𝐴_𝑖); (3.18)
(ii) the positivity of (𝑡₁𝐼−∑^𝑁_𝑖=1𝐴_𝑖)⁻¹,(𝑡₂𝐼−∑^𝑁_𝑖=1𝐴_𝑖)⁻¹ can be implied by Theorem 2.1. The right-hand inequality can be obtained by the following formula:(𝑡₂𝐼−^𝑁_𝑖=1𝐴_𝑖)⁻¹−(𝑡₁𝐼−^𝑁_𝑖=1𝐴_𝑖)⁻¹=−(𝑡₂−𝑡₁)(𝑡₁𝐼−^𝑁_𝑖=1𝐴_𝑖)⁻¹(𝑡₂𝐼−^𝑁_𝑖=1𝐴_𝑖)⁻¹. (3.19)This completes the proof.

It is important to note from above lemma that under positivity assumptions, system (2.6) is strongly stable if and only if 𝑟(𝐴₁+⋯+𝐴_𝑁)<1. Lemma 3.4. Suppose that system (2.6) is positive and strongly stable. Then, for any 𝐷∈ℝ₊^𝑛×𝑝,𝐸∈ℝ₊^𝑞×𝑛,𝜆∈ℂ⧵ℂ₁,𝑧∈(𝜕ℂ₁)^𝑁, we have‖𝐸(𝜆𝐼−^𝑁_𝑖=1𝑧_𝑖𝐴_𝑖)⁻¹𝐷‖≤‖𝐸(𝐼−^𝑁_𝑖=1𝐴_𝑖)⁻¹𝐷‖. (3.20)Proof. For any 𝜆∈ℂ⧵ℂ₁,𝑧∈(𝜕ℂ₁)^𝑁, we have 𝑟(∑^𝑁_𝑖=1𝑧_𝑖𝐴_𝑖)<1≤|𝜆|. Thus, for an arbitrary vector 𝑥∈ℝ^𝑝,|𝐸(𝜆𝐼−^𝑁_𝑖=1𝑧_𝑖𝐴_𝑖)⁻¹𝐷𝑥|≤𝐸|(𝜆𝐼−^𝑁_𝑖=1𝑧_𝑖𝐴_𝑖)⁻¹1|𝐷|𝑥|≤𝐸(|𝜆|^∞_𝑛=0|∑^𝑁_𝑖=1𝑧_𝑖𝐴_𝑖𝜆|^𝑛1)𝐷|𝑥|≤𝐸(|𝜆|^∞_𝑛=0(∑^𝑁_𝑖=1|𝑧_𝑖𝐴_𝑖|)|𝜆|^𝑛)𝐷|𝑥|≤𝐸(^∞_𝑛=0(^𝑁_𝑖=1𝐴_𝑖)^𝑛)𝐷|𝑥|≤𝐸(𝐼−^𝑁_𝑖=1𝐴_𝑖)⁻¹𝐷|𝑥|. (3.21)By Lemma 3.3, we have (𝐼−∑^𝑁_𝑖=1𝐴_𝑖)⁻¹≥0. Thus, we imply‖𝐸(𝜆𝐼−^𝑁_𝑖=1𝑧_𝑖𝐴_𝑖)⁻¹𝐷‖≤‖𝐸(𝐼−^𝑁_𝑖=1𝐴_𝑖)⁻¹𝐷‖. (3.22)Theorem 3.5. Let system (2.6) be strongly stable and positive. Assume that all 𝐷_𝑖,𝐸_𝑖,𝑖∈𝑁 are nonnegative matrices. If 𝐷_𝑖=𝐷_𝑗 or 𝐸_𝑖=𝐸_𝑗,∀𝑖,𝑗∈𝑁, then𝑟_ℂ=𝑟_ℝ=𝑟₊=1max_𝑖∈𝑁‖𝐺_𝑖𝑖(1,1)‖, (3.23)where 𝐺_𝑖𝑖(1,1)=𝐸_𝑖(𝐼−∑^𝑁_𝑖=1𝐴_𝑖)⁻¹𝐷_𝑖.Proof. By Theorem 3.2, we have𝑟_ℂ=1max_𝑖∈𝑁sup_{|𝜆|≥1,𝑧∈(𝜕ℂ1)𝑁}‖𝐺_𝑖𝑖(𝜆,𝑧)‖. (3.24)Moreover, using Lemma 3.4, we get𝑟_ℂ=1max_𝑖∈𝑁‖𝐺_𝑖𝑖(1,1)‖. (3.25)Since 𝑟_ℂ≤𝑟_ℝ≤𝑟₊, we only need to prove that𝑟₊≤1max_𝑖∈𝑁‖𝐺_𝑖𝑖(1,1)‖. (3.26)Indeed, for any 𝑖∈𝑁, since 𝐺_𝑖𝑖(1,1) is a nonnegative matrix, there exists nonnegative vector 𝑥∈ℝ^𝑝𝑖₊ such that ‖𝑥‖=1 and ‖𝐺_𝑖𝑖(1,1)𝑥‖=‖𝐺_𝑖𝑖(1,1)‖. Using Krein-Rutman theorem, see [14], there exists 𝑦^∗∈(ℝ^𝑞𝑖₊)^∗ satisfying ‖𝑦^∗‖=1 and 𝑦^∗(𝐺_𝑖𝑖(1,1)𝑥)=‖𝐺_𝑖𝑖(1,1)𝑥‖. We define a nonnegative matrix Δ∈ℝ^{𝑝𝑖×𝑞𝑖}₊ by setting1Δ=‖𝐺_𝑖𝑖(1,1)‖𝑥𝑦^∗∈ℝ^{𝑝𝑖×𝑞𝑖}₊. (3.27)Now we construct the positive disturbance Δ=(Δ₁,…,Δ_𝑁) defined by Δ_𝑗=Δ,𝑗=𝑖,0,𝑗≠𝑖. (3.28) It is easy to check that ∑^𝑁_𝑖=1‖Δ_𝑖‖=‖Δ‖=1/‖𝐺_𝑖𝑖(1,1)‖. Moreover, we have𝜆𝑥=(^𝑁_𝑖=1(𝐴_𝑖+𝐷_𝑖Δ_𝑖𝐸_𝑖))𝑥, (3.29)where 𝑥=𝐻(1,1)𝐷_𝑖𝑥. It means that Δ is a destabilizing disturbance. Thus𝑟₊≤1max_𝑖∈𝑁‖𝐺_𝑖𝑖(1,1)‖. (3.30)The proof is complete.

Now we turn to a different perturbation structure and assume that each matrix 𝐴_𝑖 is subjected to perturbations of the form𝐴_𝑖→𝐴_𝑖+^𝐾_𝑗=1𝛿_𝑖𝑗𝐵_𝑖𝑗, (3.31)where 𝐵_𝑖𝑗 are given matrices defining the structure of the perturbations and 𝛿_𝑖𝑗 are unknown scalars representing parameter uncertainties. So we can write the perturbed system𝑦(𝑡)−^𝑁_𝑖=1(𝐴_𝑖+^𝐾_𝑗=1𝛿_𝑖𝑗𝐵_𝑖𝑗)𝑦(𝑡−𝑟_𝑖)=0. (3.32)Definition 3.6. Let system (2.6) be strongly stable. The complex, real, and positive strong stability radii of system (2.6) under perturbations of the form (3.31) are defined by𝑟^𝛿_ℂ=inf{‖𝛿‖_∞∶𝛿=(𝛿_𝑖𝑗)∈ℂ^𝑁𝐾𝑟,system(3.32)isnotstronglystable},_𝛿ℝ=inf{‖𝛿‖_∞∶𝛿=(𝛿_𝑖𝑗)∈ℝ^𝑁𝐾𝑟,system(3.32)isnotstronglystable},_𝛿+=inf{‖𝛿‖_∞∶𝛿=(𝛿_𝑖𝑗)∈ℝ₊^𝑁𝐾,system(3.32)isnotstronglystable}, (3.33)respectively, we set inf∅=+∞, and ‖𝛿‖_∞=max{|𝛿_𝑖𝑗|∶𝑖∈𝑁,𝑗∈𝐾}, where 𝐾={1,…,𝐾}.Lemma 3.7. Suppose system (2.6) is strongly stable, positive and 𝐵_𝑖𝑗∈ℝ₊^𝑛×𝑛,𝑖∈𝑁,𝑗∈𝐾. Then𝑟^𝛿_ℂ=𝑟^𝛿_ℝ=𝑟^𝛿₊. (3.34)Proof. Because 𝑟^𝛿_ℂ≤𝑟^𝛿_ℝ≤𝑟^𝛿₊, we only need to prove that 𝑟^𝛿_ℂ≥𝑟^𝛿₊. Indeed, for a destabilizing disturbance 𝛿=(𝛿_𝑖𝑗)_{𝑖∈𝑁,𝑗∈𝐾}, there exist a 𝜆∈ℂ⧵ℂ₁,𝑧∈(𝜕ℂ₁)^𝑁 and a nonzero vector 𝑥∈𝕂^𝑛 such that^𝑁_𝑖=1𝑧_𝑖(𝐴_𝑖+^𝐾_𝑗=1𝛿_𝑖𝑗𝐵_𝑖𝑗)𝑥=𝜆𝑥. (3.35)This yields^𝑁_𝑖=1(𝐴_𝑖+^𝐾_𝑗=1|𝛿_𝑖𝑗|𝐵_𝑖𝑗)|𝑥|≥|^𝑁_𝑖=1𝑧_𝑖(𝐴_𝑖+^𝐾_𝑗=1𝛿_𝑖𝑗𝐵_𝑖𝑗)𝑥|=|𝜆‖𝑥|. (3.36)By Theorem 2.1, we get𝑟(^𝑁_𝑖=1(𝐴_𝑖+^𝐾_𝑗=1|𝛿_𝑖𝑗|𝐵_𝑖𝑗))≥1. (3.37)It means that |𝛿|=(|𝛿_𝑖𝑗|)_{𝑖∈𝑁,𝑗∈𝐾} is also a destabilizing disturbance. Thus, by the definition of complex and real radii, 𝑟^𝛿_ℂ≥𝑟^𝛿₊. The proof is complete.Theorem 3.8. Suppose system (2.6) is strongly stable, positive and 𝐵_𝑖𝑗∈ℝ₊^𝑛×𝑛,𝑖∈𝑁,𝑗∈𝐾. Then𝑟^𝛿_ℂ=𝑟^𝛿_ℝ=𝑟^𝛿₊=1𝑟((𝐼−∑^𝑁_𝑖=1𝐴_𝑖)⁻¹𝐵), (3.38)where 𝐵=∑_𝑖,𝑗𝐵_𝑖𝑗.Proof. By Lemma 3.7, we only need to prove that𝑟^𝛿₊=1𝑟((𝐼−∑^𝑁_𝑖=1𝐴_𝑖)⁻¹𝐵). (3.39)To do it, taking arbitrary destabilizing disturbance 𝛿=(𝛿_𝑖𝑗)_{𝑖∈𝑁,𝑗∈𝐾}∈ℝ₊^𝑁𝐾, by Lemma 3.3 and Theorem 2.1, there exist a 𝜆≥1 and a nonzero vector 𝑥∈ℝ^𝑛₊ such that(^𝑁_𝑖=1(𝐴_𝑖+^𝐾_𝑗=1𝛿_𝑖𝑗𝐵_𝑖𝑗))𝑥=𝜆𝑥, (3.40)or equivalently,_𝑖,𝑗𝛿_𝑖𝑗𝐵_𝑖𝑗𝑥=(𝜆𝐼−^𝑁_𝑖=1𝐴_𝑖)𝑥. (3.41)This yields(𝜆𝐼−^𝑁_𝑖=1𝐴_𝑖)⁻¹(_𝑖,𝑗𝛿_𝑖𝑗𝐵_𝑖𝑗)𝑥=𝑥. (3.42)Then, we have‖𝛿‖_∞(𝜆𝐼−^𝑁_𝑖=1𝐴_𝑖)⁻¹𝐵𝑥≥(𝜆𝐼−^𝑁_𝑖=1𝐴_𝑖)⁻¹(_𝑖,𝑗𝛿_𝑖𝑗𝐵_𝑖𝑗)𝑥=𝑥. (3.43)Using Theorem 2.1 again, we obtain𝑟((𝜆𝐼−^𝑁_𝑖=1𝐴_𝑖)⁻¹1𝐵)≥‖𝛿‖_∞ (3.44)or equivalently,‖𝛿‖_∞≥1𝑟((𝜆𝐼−∑^𝑁_𝑖=1𝐴_𝑖)⁻¹𝐵). (3.45)Thus, from the definition of 𝑟^𝛿₊, one has𝑟^𝛿₊≥1𝑟((𝐼−∑^𝑁_𝑖=1𝐴_𝑖)⁻¹𝐵). (3.46)On the other hand, setting 𝜆=𝑟((𝐼−∑^𝑁_𝑖=1𝐴_𝑖)⁻¹𝐵)>0. Then, by Theorem 2.1, there exists a nonnegative vector 𝑥∈ℝ^𝑛₊ satisfying((𝐼−^𝑁_𝑖=1𝐴_𝑖)⁻¹𝐵)𝑥=𝜆𝑥. (3.47)This is equivalent to𝐵𝑥=𝜆(𝐼−^𝑁_𝑖=1𝐴_𝑖)𝑥. (3.48)Hence,^𝑁_𝑖=1(𝐴_𝑖+1𝜆𝐵)𝑥=𝑥. (3.49)This means that 𝛿=(1/𝜆)∈ℝ₊^𝑁𝐾 is a destabilizing disturbance and thus, 𝑟^𝛿₊≤1/𝑟((𝐼−∑^𝑁_𝑖=1𝐴_𝑖)⁻¹𝐵). The proof is complete.

Now we consider the following example to illustrate the obtained results.

Example 3.9. Consider system𝑦(𝑡)=𝐴₁𝑦(𝑡−𝑟)+𝐴₂𝑦(𝑡−𝑠), (3.50)where𝐴₁=1413140,𝐴₂=0161413. (3.51)Then we have𝑟(𝐴₁+𝐴₂√)=7+14524<1. (3.52)Thus system (3.50) is strongly stable.

Assume that the matrices 𝐴₁,𝐴₂ are subjected to perturbations of the form 𝐴₁→𝐴₁+𝐷₁Δ₁𝐸₁,𝐴₂→𝐴₂+𝐷₂Δ₂𝐸₂, where𝐷₁=𝐷₂=1010,𝐸₁=1100,𝐸₂=1010. (3.53)Then𝐺₁₁(1,1)=293000,𝐺₂₂(1,1)=143500. (3.54)If ℝ² is provided with the norm defined by ‖(𝑥,𝑦)‖=|𝑥|+|𝑦|, then by Theorem 3.5, we have𝑟_ℂ=𝑟_ℝ=𝑟₊=329. (3.55)Assume that the given two matrices 𝐴₁,𝐴₂ are subjected to perturbations of the form 𝐴₁→𝐴₁+𝛿₁₁𝐵₁₁+𝛿₁₂𝐵₁₂,𝐴₂→𝐴₂+𝛿₂₁𝐵₂₁+𝛿₂₂𝐵₂₂, where𝐵₁₁=12101,𝐵₁₂=113112,𝐵₂₁=101314,𝐵₂₂=121311. (3.56)Then(𝐼−𝐴₁−𝐴₂)⁻¹𝐵=10227181116312. (3.57)By Theorem 3.8, we get𝑟^𝛿_ℂ=𝑟^𝛿_ℝ=𝑟^𝛿₊=24283+√81753. (3.58)