# The Robustness of Strong Stability of   Positive Homogeneous Difference Equations

Authors: The Anh Bui; Dang Xuan Thanh Duong
Journal: Journal of Applied Mathematics
 (2008)

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## 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.


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## Body

## 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 Section2, 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. Letn,⁢l,⁢q be positive integers, a matrix P=[pij]∈ℝl×q is said to be nonnegative (P≥0) if all its entries pij are nonnegative. It is said to be positive (P>0) if all its entries pij are positive. For P,Q∈ℝl×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∈𝕂n and P∈𝕂l×q,
we define |x|∈ℝ+n and |P|∈ℝ+l×q by |x|=(|xi|),|P|=[|pij|].
For any matrix A∈𝕂n×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 𝕂n is said to be monotonic if it satisfies(2.1)|x|≤|y|⇒∥x∥≤∥y∥,∀x,y∈𝕂n.It can be shown that a vector
norm ∥⋅∥ on 𝕂n is monotonic if and only if ∥x∥=∥|x|∥ for all x∈𝕂n,
see [7]. All norms on 𝕂n we use in this paper are assumed to be
monotonic.
Theorem 2.1 (Perron-Frobenius).
Suppose thatA∈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)
Ifλ∈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)
Givenα>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.
LetA∈𝕂n×n,⁢B∈ℝ+n×n.
If |A|≤B then(2.2)r(A)≤r(|A|)≤r(B).


### 2.2. Homogeneous Difference Equations

Consider the
neutral differential difference equation of the following form:(2.3)ddt[D(r,A)yt]=f(t,yt),where D(r,A):C([−h;0],ℝn)→ℝn is linear continuous defined
by(2.4)D(r,A)ϕ=ϕ(0)−∑i=1NAiϕ(−ri),ϕ∈C([−h;0],ℝn).Here each Ai is an n×n-matrix, each ri is a constant satisfying ri>0 and h=max⁡{ri:i∈N¯},N¯={1,2,…,N} and yt∈C([−h;0],ℝn) is defined by yt(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
equations(2.5)D(r,A)yt=0,t≥0,or equivalently,(2.6)y(t)−∑i=1NAiy(t−ri)=0.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 operatorD(r,A) or system (2.6) is called stable if the zero
solution of (2.6) with y0∈CD(r,A)={ϕ∈C([−h,0],ℝn):D(r,A)ϕ=0} is uniformly asymptotically stable.Associated with system (2.6) we define the
quasipolynomial(2.7)H(s)=I−∑k=1Ne−srkAk.For s∈ℂ,
if det⁡H(s)=0, then s is called a characteristic root of the
quasipolynomial matrix (2.7). Then, a nonzero vector x∈ℂn 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 aH=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 aH<0.It is well known that aH is not continuous in the delays (r1,…,rN),
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 (ri)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(∑i=1NziAi):|zi|=1,i∈N¯}<1.We set ℂ1={z∈ℂ:|z|<1} and ∂ℂ1={z∈ℂ:|z|=1}.
Since r(⋅) is continuous in ℂn×n,
we imply the continuity of the following function g:(∂ℂ1)N→ℝ defined by(2.8)g(z1,…,zn)=r(∑i=1NziAi).Moreover, by the compactness of
the set (∂ℂ1)N,
there exists z*=(z1*,…,zN*) such that(2.9)r(∑i=1Nzi*Ai)=sup⁡{r(∑i=1NziAi):|zi|=1,i∈N¯}.By the above result, we can get
the following statement: system (2.6) is strongly stable if and only
if(2.10)max⁡{r(∑i=1NziAi):(zi)i∈N¯∈(∂ℂ1)N}<1.


## 2.1. Nonnegative Matrices

We first
introduce some notations. Letn,⁢l,⁢q be positive integers, a matrix P=[pij]∈ℝl×q is said to be nonnegative (P≥0) if all its entries pij are nonnegative. It is said to be positive (P>0) if all its entries pij are positive. For P,Q∈ℝl×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∈𝕂n and P∈𝕂l×q,
we define |x|∈ℝ+n and |P|∈ℝ+l×q by |x|=(|xi|),|P|=[|pij|].
For any matrix A∈𝕂n×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 𝕂n is said to be monotonic if it satisfies(2.1)|x|≤|y|⇒∥x∥≤∥y∥,∀x,y∈𝕂n.It can be shown that a vector
norm ∥⋅∥ on 𝕂n is monotonic if and only if ∥x∥=∥|x|∥ for all x∈𝕂n,
see [7]. All norms on 𝕂n we use in this paper are assumed to be
monotonic.
Theorem 2.1 (Perron-Frobenius).
Suppose thatA∈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)
Ifλ∈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)
Givenα>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.
LetA∈𝕂n×n,⁢B∈ℝ+n×n.
If |A|≤B then(2.2)r(A)≤r(|A|)≤r(B).


## 2.2. Homogeneous Difference Equations

Consider the
neutral differential difference equation of the following form:(2.3)ddt[D(r,A)yt]=f(t,yt),where D(r,A):C([−h;0],ℝn)→ℝn is linear continuous defined
by(2.4)D(r,A)ϕ=ϕ(0)−∑i=1NAiϕ(−ri),ϕ∈C([−h;0],ℝn).Here each Ai is an n×n-matrix, each ri is a constant satisfying ri>0 and h=max⁡{ri:i∈N¯},N¯={1,2,…,N} and yt∈C([−h;0],ℝn) is defined by yt(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
equations(2.5)D(r,A)yt=0,t≥0,or equivalently,(2.6)y(t)−∑i=1NAiy(t−ri)=0.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 operatorD(r,A) or system (2.6) is called stable if the zero
solution of (2.6) with y0∈CD(r,A)={ϕ∈C([−h,0],ℝn):D(r,A)ϕ=0} is uniformly asymptotically stable.Associated with system (2.6) we define the
quasipolynomial(2.7)H(s)=I−∑k=1Ne−srkAk.For s∈ℂ,
if det⁡H(s)=0, then s is called a characteristic root of the
quasipolynomial matrix (2.7). Then, a nonzero vector x∈ℂn 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 aH=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 aH<0.It is well known that aH is not continuous in the delays (r1,…,rN),
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 (ri)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(∑i=1NziAi):|zi|=1,i∈N¯}<1.We set ℂ1={z∈ℂ:|z|<1} and ∂ℂ1={z∈ℂ:|z|=1}.
Since r(⋅) is continuous in ℂn×n,
we imply the continuity of the following function g:(∂ℂ1)N→ℝ defined by(2.8)g(z1,…,zn)=r(∑i=1NziAi).Moreover, by the compactness of
the set (∂ℂ1)N,
there exists z*=(z1*,…,zN*) such that(2.9)r(∑i=1Nzi*Ai)=sup⁡{r(∑i=1NziAi):|zi|=1,i∈N¯}.By the above result, we can get
the following statement: system (2.6) is strongly stable if and only
if(2.10)max⁡{r(∑i=1NziAi):(zi)i∈N¯∈(∂ℂ1)N}<1.


## 3. Main Results

### 3.1. Complex Strong Stability Radius

Suppose that
system (2.6) is strongly stable. Now we assume that each matrix Ai is subjected to the perturbation of the
form(3.1)Ai→Ai+DiΔiEi,i∈N¯,where Di∈ℂn×pi,⁢Ei∈ℂqi×n are given matrices defining the structure of
the perturbations and Δi∈ℂpi×qi are unknown matrices, i∈N¯.
We write the perturbed system(3.2)y(t)−∑i=1N(Ai+DiΔiEi)y(t−ri)=0.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
by(3.3)rℂ=inf⁡{∑i=1N∥Δi∥:Δi∈ℂpi×qi,⁢i∈N¯,system(3.2)isnotstronglystable},rℝ=inf⁡{∑i=1N∥Δi∥:Δi∈ℝpi×qi,⁢i∈N¯,system(3.2)isnotstronglystable},r+=inf⁡{∑i=1N∥Δi∥:Δi∈ℝ+pi×qi,⁢i∈N¯,system(3.2)isnotstronglystable},respectively, we set inf⁡∅=+∞.If system (2.6) is strongly stable, we define a
function H(⋅,⋅):ℂ∖ℂ1×(∂ℂ1)N→ℂn×n by H(λ,z)=(λI−∑i=1NziAi)−1,⁢λ∈ℂ∖ℂ1,z=(zi)i∈N¯∈(∂ℂ1)N.
It is easy to see that H(⋅,⋅) is well-defined. For any λ∈ℂ∖ℂ1,z=(zi)i∈N¯∈(∂ℂ1)N,
we set(3.4)Gij(λ,z)=EiH(λ,z)Dj∈ℂqi×pj.Theorem 3.2.
Let system (2.6) be strongly stable. Then we have
(i)
(3.5)1max⁡i,j∈N¯⁢sup⁡|λ|≥1,z∈(∂ℂ1)N∥Gij(λ,z)∥≤rℂ≤1max⁡i∈N¯⁢sup⁡|λ|≥1,z∈(∂ℂ1)N∥Gii(λ,z)∥,(ii)
in
particular, ifDi=Dj (or Ei=Ej) for all i,j∈N¯,
then we have(3.6)rℂ=1max⁡i∈N¯⁢sup⁡|λ|≥1,z∈(∂ℂ1)N∥Gii(λ,z)∥.Proof.
LetΔ=(Δi)i∈N¯ be a destabilizing disturbance. Then there
exists (λ,z)∈ℂ∖ℂ1×(∂ℂ1)N such that λ∈σ(∑i=1Nzi(Ai+DiΔiEi)).
This means that there exists a nonzero vector x satisfying(3.7)(∑i=1Nzi(Ai+DiΔiEi))x=λx.This follows
that(3.8)(∑i=1NziDiΔiEi)x=(λI−∑i=1NziAi)x,or equivalently,(3.9)x=∑i=1NziH(λ,z)DiΔiEix.Choose q∈N¯ such that ∥Eqx∥=max⁡{∥Eix∥:i∈N¯}.
Multiplying the above equation with Eq,
we obtain(3.10)Eqx=∑i=1NziEqH(λ,z)DiΔiEix=∑i=1NziGqi(λ,z)ΔiEix.This implies
that(3.11)∥Eqx∥≤∑i=1N∥zi∥∥Gqi(λ,z)∥∥Δi∥∥Eix∥≤∑i=1Nmax⁡i,j∈N¯⁢sup⁡|λ|≥1,z∈(∂ℂ1)N∥Gij(λ,z)∥∥Δi∥∥Eqx∥.From this inequality and the
definition of rℂ,
the left-hand inequality of (i) follows:(3.12)rℂ≥1max⁡i,j∈N¯⁢sup⁡|λ|≥1,z∈(∂ℂ1)N∥Gij(λ,z)∥.Now it remains to prove the
right-hand inequality of (i):  (3.13)rℂ≤1max⁡i∈N¯⁢sup⁡|λ|≥1,z∈(∂ℂ1)N∥Gii(λ,z)∥.Indeed, for any (λ,z)∈ℂ∖ℂ1×(∂ℂ1)N,
and i∈N¯,
there exists nonzero vector x∈ℂpi such that ∥x∥=1 and ∥Gii(λ,z)x∥=∥Gii(λ,z)∥.
By Hahn-Banach theorem, there exists y*∈(ℂqi)* satisfying ∥y*∥=1 and y*(Gii(λ,z)x)=∥Gii(λ,z)x∥.
We define a matrix Δ¯∈ℂpi×qi by setting(3.14)Δ¯=1∥Gii(λ,z)∥xy*∈ℂpi×qi.Now we construct the disturbance Δ=(Δ1,…,ΔN) defined by (3.15)Δj={zi−1Δ¯,j=i,0,j⁢≠i. It is easy to check that ∑i=1N∥Δi∥=∥Δ¯∥=1/∥Gii(λ,z)∥.
Moreover, we have(3.16)λx^=(∑i=1Nzi(Ai+DiΔiEi))x,^where x^=H(λ,z)Dix.
This means that Δ is a destabilizing disturbance.
Thus,(3.17)rℂ≤1max⁡i∈N¯⁢sup⁡|λ|≥1,z∈(∂ℂ1)N∥Gii(λ,z)∥.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, Ai are nonnegative for all i∈N¯.Lemma 3.3.
LetAi∈ℝ+n×n.
Then we have
(i)
r(∑i=1NAi)=sup⁡z∈(∂ℂ1)Nr(∑i=1NziAi);(ii)
r(∑i=1NAi)<t1≤t2⇒0≤(t2I−∑i=1NAi)−1≤(t1I−∑i=1NAi)−1,Proof.
(i) By Theorem2.2, we have(3.18)r(∑i=1NziAi)≤r(∑i=1N|ziAi|)=r(∑i=1NAi);
(ii) the positivity of(t1I−∑i=1NAi)−1,(t2I−∑i=1NAi)−1 can be implied by Theorem 2.1. The right-hand
inequality can be obtained by the following formula:(3.19)(t2I−∑i=1NAi)−1−(t1I−∑i=1NAi)−1=−(t2−t1)(t1I−∑i=1NAi)−1(t2I−∑i=1NAi)−1.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+⋯+AN)<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(3.20)∥E(λI−∑i=1NziAi)−1D∥≤∥E(I−∑i=1NAi)−1D∥.Proof.
For anyλ∈ℂ∖ℂ1,z∈(∂ℂ1)N,
we have r(∑i=1NziAi)<1≤|λ|.
Thus, for an arbitrary vector x∈ℝp,(3.21)|E(λI−∑i=1NziAi)−1Dx|≤E|(λI−∑i=1NziAi)−1|D|x|≤E1|λ|(∑n=0∞|∑i=1NziAiλ|n)D|x|≤E1|λ|(∑n=0∞(∑i=1N|ziAi||λ|)n)D|x|≤E(∑n=0∞(∑i=1NAi)n)D|x|≤E(I−∑i=1NAi)−1D|x|.By Lemma 3.3, we have (I−∑i=1NAi)−1≥0.
Thus, we imply(3.22)∥E(λI−∑i=1NziAi)−1D∥≤∥E(I−∑i=1NAi)−1D∥.Theorem 3.5.
Let system (2.6) be strongly stable and positive. Assume
that all Di,⁢Ei,⁢i∈N¯ are nonnegative matrices. If Di=Dj or Ei=Ej,⁢∀i,j∈N¯,
then(3.23)rℂ=rℝ=r+=1max⁡i∈N¯∥Gii(1,1)∥,where Gii(1,1)=Ei(I−∑i=1NAi)−1Di.Proof.
By Theorem3.2, we
have(3.24)rℂ=1max⁡i∈N¯⁢sup⁡|λ|≥1,z∈(∂ℂ1)N∥Gii(λ,z)∥.Moreover, using Lemma 3.4, we
get(3.25)rℂ=1max⁡i∈N¯∥Gii(1,1)∥.Since rℂ≤rℝ≤r+,
we only need to prove that(3.26)r+≤1max⁡i∈N¯∥Gii(1,1)∥.Indeed, for any i∈N¯,
since Gii(1,1) is a nonnegative matrix, there exists
nonnegative vector x∈ℝ+pi such that ∥x∥=1 and ∥Gii(1,1)x∥=∥Gii(1,1)∥.
Using Krein-Rutman theorem, see [14], there exists y*∈(ℝ+qi)* satisfying ∥y*∥=1 and y*(Gii(1,1)x)=∥Gii(1,1)x∥.
We define a nonnegative matrix Δ¯∈ℝ+pi×qi by setting(3.27)Δ¯=1∥Gii(1,1)∥xy*∈ℝ+pi×qi.Now we construct the positive
disturbance Δ=(Δ1,…,ΔN) defined by (3.28)Δj={Δ¯,j=i,0,j≠i. It is easy to check that ∑i=1N∥Δi∥=∥Δ¯∥=1/∥Gii(1,1)∥.
Moreover, we have(3.29)λx^=(∑i=1N(Ai+DiΔiEi))x^,where x^=H(1,1)Dix.
It means that Δ is a destabilizing disturbance.
Thus(3.30)r+≤1max⁡i∈N¯∥Gii(1,1)∥.The proof is complete.Now we turn to a different perturbation structure and
assume that each matrixAi is subjected to perturbations of the
form(3.31)Ai→Ai+∑j=1KδijBij,where Bij are given matrices defining the structure of
the perturbations and δij are unknown scalars representing parameter
uncertainties. So we can write the perturbed system(3.32)y(t)−∑i=1N(Ai+∑j=1KδijBij)y(t−ri)=0.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
by(3.33)rℂδ=inf⁡{∥δ∥∞:δ=(δij)∈ℂNK,system(3.32)isnotstronglystable},rℝδ=inf⁡{∥δ∥∞:δ=(δij)∈ℝNK,system(3.32)isnotstronglystable},r+δ=inf⁡{∥δ∥∞:δ=(δij)∈ℝ+NK,system(3.32)isnotstronglystable},respectively, we set inf⁡∅=+∞,
and ∥δ∥∞=max⁡{|δij|:i∈N¯,j∈K¯},
where K¯={1,…,K}.Lemma 3.7.
Suppose system (2.6) is strongly
stable, positive and Bij∈ℝ+n×n,⁢i∈N¯,j∈K¯.
Then(3.34)rℂδ=rℝδ=r+δ.Proof.
Becauserℂδ≤rℝδ≤r+δ,
we only need to prove that rℂδ≥r+δ. Indeed, for a destabilizing disturbance δ=(δij)i∈N¯,j∈K¯,
there exist a λ∈ℂ∖ℂ1,z∈(∂ℂ1)N and a nonzero vector x∈𝕂n such that(3.35)∑i=1Nzi(Ai+∑j=1KδijBij)x=λx.This yields(3.36)∑i=1N(Ai+∑j=1K|δij|Bij)|x|≥|∑i=1Nzi(Ai+∑j=1KδijBij)x|=|λ∥x|.By Theorem 2.1, we
get(3.37)r(∑i=1N(Ai+∑j=1K|δij|Bij))≥1.It means that |δ|=(|δij|)i∈N¯,j∈K¯ is also a destabilizing disturbance. Thus, by
the definition of complex and real radii, rℂδ≥r+δ.
The proof is complete.Theorem 3.8.
Suppose system (2.6) is strongly stable, positive and Bij∈ℝ+n×n,⁢i∈N¯,⁢j∈K¯.
Then(3.38)rℂδ=rℝδ=r+δ=1r((I−∑i=1NAi)−1B),where B=∑i,jBij.Proof.
By Lemma3.7, we only need to prove
that(3.39)r+δ=1r((I−∑i=1NAi)−1B).To do it, taking arbitrary
destabilizing disturbance δ=(δij)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(3.40)(∑i=1N(Ai+∑j=1KδijBij))x=λx,or equivalently,(3.41)∑i,jδijBijx=(λI−∑i=1NAi)x.This yields(3.42)(λI−∑i=1NAi)−1(∑i,jδijBij)x=x.Then, we have(3.43)∥δ∥∞(λI−∑i=1NAi)−1Bx≥(λI−∑i=1NAi)−1(∑i,jδijBij)x=x.Using Theorem 2.1 again, we
obtain(3.44)r((λI−∑i=1NAi)−1B)≥1∥δ∥∞or equivalently,(3.45)∥δ∥∞≥1r((λI−∑i=1NAi)−1B).Thus, from the definition of r+δ,
one has(3.46)r+δ≥1r((I−∑i=1NAi)−1B).On the other hand, setting λ=r((I−∑i=1NAi)−1B)>0.
Then, by Theorem 2.1, there exists a nonnegative vector x∈ℝ+n satisfying(3.47)((I−∑i=1NAi)−1B)x=λx.This is equivalent
to(3.48)Bx=λ(I−∑i=1NAi)x.Hence,(3.49)∑i=1N(Ai+1λB)x=x.This means that δ=(1/λ)∈ℝ+NK is a destabilizing disturbance and thus, r+δ≤1/r((I−∑i=1NAi)−1B).
The proof is complete.Now we consider the following example to illustrate
the obtained results.Example 3.9.
Consider system(3.50)y(t)=A1y(t−r)+A2y(t−s),where(3.51)A1=(1413140),A2=(0161413).Then we have(3.52)r(A1+A2)=7+14524<1.Thus system (3.50) is strongly
stable.Assume that the matricesA1,A2 are subjected to perturbations of the form A1→A1+D1Δ1E1,⁢⁢A2→A2+D2Δ2E2,
where(3.53)D1=D2=(1010),E1=(1100),E2=(1010).Then(3.54)G11(1,1)=(293000),G22(1,1)=(143500).If ℝ2 is provided with the norm defined by ∥(x,y)∥=|x|+|y|,
then by Theorem 3.5, we have(3.55)rℂ=rℝ=r+=329.Assume that the given two
matrices A1,⁢A2 are subjected to perturbations of the form A1→A1+δ11B11+δ12B12,⁢A2→A2+δ21B21+δ22B22,
where(3.56)B11=(12101),B12=(113112),B21=(011314),B22=(121311).Then(3.57)(I−A1−A2)−1B=(10227181116312).By Theorem 3.8, we get(3.58)rℂδ=rℝδ=r+δ=24283+81753.


## 3.1. Complex Strong Stability Radius

Suppose that
system (2.6) is strongly stable. Now we assume that each matrix Ai is subjected to the perturbation of the
form(3.1)Ai→Ai+DiΔiEi,i∈N¯,where Di∈ℂn×pi,⁢Ei∈ℂqi×n are given matrices defining the structure of
the perturbations and Δi∈ℂpi×qi are unknown matrices, i∈N¯.
We write the perturbed system(3.2)y(t)−∑i=1N(Ai+DiΔiEi)y(t−ri)=0.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
by(3.3)rℂ=inf⁡{∑i=1N∥Δi∥:Δi∈ℂpi×qi,⁢i∈N¯,system(3.2)isnotstronglystable},rℝ=inf⁡{∑i=1N∥Δi∥:Δi∈ℝpi×qi,⁢i∈N¯,system(3.2)isnotstronglystable},r+=inf⁡{∑i=1N∥Δi∥:Δi∈ℝ+pi×qi,⁢i∈N¯,system(3.2)isnotstronglystable},respectively, we set inf⁡∅=+∞.If system (2.6) is strongly stable, we define a
function H(⋅,⋅):ℂ∖ℂ1×(∂ℂ1)N→ℂn×n by H(λ,z)=(λI−∑i=1NziAi)−1,⁢λ∈ℂ∖ℂ1,z=(zi)i∈N¯∈(∂ℂ1)N.
It is easy to see that H(⋅,⋅) is well-defined. For any λ∈ℂ∖ℂ1,z=(zi)i∈N¯∈(∂ℂ1)N,
we set(3.4)Gij(λ,z)=EiH(λ,z)Dj∈ℂqi×pj.Theorem 3.2.
Let system (2.6) be strongly stable. Then we have
(i)
(3.5)1max⁡i,j∈N¯⁢sup⁡|λ|≥1,z∈(∂ℂ1)N∥Gij(λ,z)∥≤rℂ≤1max⁡i∈N¯⁢sup⁡|λ|≥1,z∈(∂ℂ1)N∥Gii(λ,z)∥,(ii)
in
particular, ifDi=Dj (or Ei=Ej) for all i,j∈N¯,
then we have(3.6)rℂ=1max⁡i∈N¯⁢sup⁡|λ|≥1,z∈(∂ℂ1)N∥Gii(λ,z)∥.Proof.
LetΔ=(Δi)i∈N¯ be a destabilizing disturbance. Then there
exists (λ,z)∈ℂ∖ℂ1×(∂ℂ1)N such that λ∈σ(∑i=1Nzi(Ai+DiΔiEi)).
This means that there exists a nonzero vector x satisfying(3.7)(∑i=1Nzi(Ai+DiΔiEi))x=λx.This follows
that(3.8)(∑i=1NziDiΔiEi)x=(λI−∑i=1NziAi)x,or equivalently,(3.9)x=∑i=1NziH(λ,z)DiΔiEix.Choose q∈N¯ such that ∥Eqx∥=max⁡{∥Eix∥:i∈N¯}.
Multiplying the above equation with Eq,
we obtain(3.10)Eqx=∑i=1NziEqH(λ,z)DiΔiEix=∑i=1NziGqi(λ,z)ΔiEix.This implies
that(3.11)∥Eqx∥≤∑i=1N∥zi∥∥Gqi(λ,z)∥∥Δi∥∥Eix∥≤∑i=1Nmax⁡i,j∈N¯⁢sup⁡|λ|≥1,z∈(∂ℂ1)N∥Gij(λ,z)∥∥Δi∥∥Eqx∥.From this inequality and the
definition of rℂ,
the left-hand inequality of (i) follows:(3.12)rℂ≥1max⁡i,j∈N¯⁢sup⁡|λ|≥1,z∈(∂ℂ1)N∥Gij(λ,z)∥.Now it remains to prove the
right-hand inequality of (i):  (3.13)rℂ≤1max⁡i∈N¯⁢sup⁡|λ|≥1,z∈(∂ℂ1)N∥Gii(λ,z)∥.Indeed, for any (λ,z)∈ℂ∖ℂ1×(∂ℂ1)N,
and i∈N¯,
there exists nonzero vector x∈ℂpi such that ∥x∥=1 and ∥Gii(λ,z)x∥=∥Gii(λ,z)∥.
By Hahn-Banach theorem, there exists y*∈(ℂqi)* satisfying ∥y*∥=1 and y*(Gii(λ,z)x)=∥Gii(λ,z)x∥.
We define a matrix Δ¯∈ℂpi×qi by setting(3.14)Δ¯=1∥Gii(λ,z)∥xy*∈ℂpi×qi.Now we construct the disturbance Δ=(Δ1,…,ΔN) defined by (3.15)Δj={zi−1Δ¯,j=i,0,j⁢≠i. It is easy to check that ∑i=1N∥Δi∥=∥Δ¯∥=1/∥Gii(λ,z)∥.
Moreover, we have(3.16)λx^=(∑i=1Nzi(Ai+DiΔiEi))x,^where x^=H(λ,z)Dix.
This means that Δ is a destabilizing disturbance.
Thus,(3.17)rℂ≤1max⁡i∈N¯⁢sup⁡|λ|≥1,z∈(∂ℂ1)N∥Gii(λ,z)∥.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, Ai are nonnegative for all i∈N¯.Lemma 3.3.
LetAi∈ℝ+n×n.
Then we have
(i)
r(∑i=1NAi)=sup⁡z∈(∂ℂ1)Nr(∑i=1NziAi);(ii)
r(∑i=1NAi)<t1≤t2⇒0≤(t2I−∑i=1NAi)−1≤(t1I−∑i=1NAi)−1,Proof.
(i) By Theorem2.2, we have(3.18)r(∑i=1NziAi)≤r(∑i=1N|ziAi|)=r(∑i=1NAi);
(ii) the positivity of(t1I−∑i=1NAi)−1,(t2I−∑i=1NAi)−1 can be implied by Theorem 2.1. The right-hand
inequality can be obtained by the following formula:(3.19)(t2I−∑i=1NAi)−1−(t1I−∑i=1NAi)−1=−(t2−t1)(t1I−∑i=1NAi)−1(t2I−∑i=1NAi)−1.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+⋯+AN)<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(3.20)∥E(λI−∑i=1NziAi)−1D∥≤∥E(I−∑i=1NAi)−1D∥.Proof.
For anyλ∈ℂ∖ℂ1,z∈(∂ℂ1)N,
we have r(∑i=1NziAi)<1≤|λ|.
Thus, for an arbitrary vector x∈ℝp,(3.21)|E(λI−∑i=1NziAi)−1Dx|≤E|(λI−∑i=1NziAi)−1|D|x|≤E1|λ|(∑n=0∞|∑i=1NziAiλ|n)D|x|≤E1|λ|(∑n=0∞(∑i=1N|ziAi||λ|)n)D|x|≤E(∑n=0∞(∑i=1NAi)n)D|x|≤E(I−∑i=1NAi)−1D|x|.By Lemma 3.3, we have (I−∑i=1NAi)−1≥0.
Thus, we imply(3.22)∥E(λI−∑i=1NziAi)−1D∥≤∥E(I−∑i=1NAi)−1D∥.Theorem 3.5.
Let system (2.6) be strongly stable and positive. Assume
that all Di,⁢Ei,⁢i∈N¯ are nonnegative matrices. If Di=Dj or Ei=Ej,⁢∀i,j∈N¯,
then(3.23)rℂ=rℝ=r+=1max⁡i∈N¯∥Gii(1,1)∥,where Gii(1,1)=Ei(I−∑i=1NAi)−1Di.Proof.
By Theorem3.2, we
have(3.24)rℂ=1max⁡i∈N¯⁢sup⁡|λ|≥1,z∈(∂ℂ1)N∥Gii(λ,z)∥.Moreover, using Lemma 3.4, we
get(3.25)rℂ=1max⁡i∈N¯∥Gii(1,1)∥.Since rℂ≤rℝ≤r+,
we only need to prove that(3.26)r+≤1max⁡i∈N¯∥Gii(1,1)∥.Indeed, for any i∈N¯,
since Gii(1,1) is a nonnegative matrix, there exists
nonnegative vector x∈ℝ+pi such that ∥x∥=1 and ∥Gii(1,1)x∥=∥Gii(1,1)∥.
Using Krein-Rutman theorem, see [14], there exists y*∈(ℝ+qi)* satisfying ∥y*∥=1 and y*(Gii(1,1)x)=∥Gii(1,1)x∥.
We define a nonnegative matrix Δ¯∈ℝ+pi×qi by setting(3.27)Δ¯=1∥Gii(1,1)∥xy*∈ℝ+pi×qi.Now we construct the positive
disturbance Δ=(Δ1,…,ΔN) defined by (3.28)Δj={Δ¯,j=i,0,j≠i. It is easy to check that ∑i=1N∥Δi∥=∥Δ¯∥=1/∥Gii(1,1)∥.
Moreover, we have(3.29)λx^=(∑i=1N(Ai+DiΔiEi))x^,where x^=H(1,1)Dix.
It means that Δ is a destabilizing disturbance.
Thus(3.30)r+≤1max⁡i∈N¯∥Gii(1,1)∥.The proof is complete.Now we turn to a different perturbation structure and
assume that each matrixAi is subjected to perturbations of the
form(3.31)Ai→Ai+∑j=1KδijBij,where Bij are given matrices defining the structure of
the perturbations and δij are unknown scalars representing parameter
uncertainties. So we can write the perturbed system(3.32)y(t)−∑i=1N(Ai+∑j=1KδijBij)y(t−ri)=0.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
by(3.33)rℂδ=inf⁡{∥δ∥∞:δ=(δij)∈ℂNK,system(3.32)isnotstronglystable},rℝδ=inf⁡{∥δ∥∞:δ=(δij)∈ℝNK,system(3.32)isnotstronglystable},r+δ=inf⁡{∥δ∥∞:δ=(δij)∈ℝ+NK,system(3.32)isnotstronglystable},respectively, we set inf⁡∅=+∞,
and ∥δ∥∞=max⁡{|δij|:i∈N¯,j∈K¯},
where K¯={1,…,K}.Lemma 3.7.
Suppose system (2.6) is strongly
stable, positive and Bij∈ℝ+n×n,⁢i∈N¯,j∈K¯.
Then(3.34)rℂδ=rℝδ=r+δ.Proof.
Becauserℂδ≤rℝδ≤r+δ,
we only need to prove that rℂδ≥r+δ. Indeed, for a destabilizing disturbance δ=(δij)i∈N¯,j∈K¯,
there exist a λ∈ℂ∖ℂ1,z∈(∂ℂ1)N and a nonzero vector x∈𝕂n such that(3.35)∑i=1Nzi(Ai+∑j=1KδijBij)x=λx.This yields(3.36)∑i=1N(Ai+∑j=1K|δij|Bij)|x|≥|∑i=1Nzi(Ai+∑j=1KδijBij)x|=|λ∥x|.By Theorem 2.1, we
get(3.37)r(∑i=1N(Ai+∑j=1K|δij|Bij))≥1.It means that |δ|=(|δij|)i∈N¯,j∈K¯ is also a destabilizing disturbance. Thus, by
the definition of complex and real radii, rℂδ≥r+δ.
The proof is complete.Theorem 3.8.
Suppose system (2.6) is strongly stable, positive and Bij∈ℝ+n×n,⁢i∈N¯,⁢j∈K¯.
Then(3.38)rℂδ=rℝδ=r+δ=1r((I−∑i=1NAi)−1B),where B=∑i,jBij.Proof.
By Lemma3.7, we only need to prove
that(3.39)r+δ=1r((I−∑i=1NAi)−1B).To do it, taking arbitrary
destabilizing disturbance δ=(δij)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(3.40)(∑i=1N(Ai+∑j=1KδijBij))x=λx,or equivalently,(3.41)∑i,jδijBijx=(λI−∑i=1NAi)x.This yields(3.42)(λI−∑i=1NAi)−1(∑i,jδijBij)x=x.Then, we have(3.43)∥δ∥∞(λI−∑i=1NAi)−1Bx≥(λI−∑i=1NAi)−1(∑i,jδijBij)x=x.Using Theorem 2.1 again, we
obtain(3.44)r((λI−∑i=1NAi)−1B)≥1∥δ∥∞or equivalently,(3.45)∥δ∥∞≥1r((λI−∑i=1NAi)−1B).Thus, from the definition of r+δ,
one has(3.46)r+δ≥1r((I−∑i=1NAi)−1B).On the other hand, setting λ=r((I−∑i=1NAi)−1B)>0.
Then, by Theorem 2.1, there exists a nonnegative vector x∈ℝ+n satisfying(3.47)((I−∑i=1NAi)−1B)x=λx.This is equivalent
to(3.48)Bx=λ(I−∑i=1NAi)x.Hence,(3.49)∑i=1N(Ai+1λB)x=x.This means that δ=(1/λ)∈ℝ+NK is a destabilizing disturbance and thus, r+δ≤1/r((I−∑i=1NAi)−1B).
The proof is complete.Now we consider the following example to illustrate
the obtained results.Example 3.9.
Consider system(3.50)y(t)=A1y(t−r)+A2y(t−s),where(3.51)A1=(1413140),A2=(0161413).Then we have(3.52)r(A1+A2)=7+14524<1.Thus system (3.50) is strongly
stable.Assume that the matricesA1,A2 are subjected to perturbations of the form A1→A1+D1Δ1E1,⁢⁢A2→A2+D2Δ2E2,
where(3.53)D1=D2=(1010),E1=(1100),E2=(1010).Then(3.54)G11(1,1)=(293000),G22(1,1)=(143500).If ℝ2 is provided with the norm defined by ∥(x,y)∥=|x|+|y|,
then by Theorem 3.5, we have(3.55)rℂ=rℝ=r+=329.Assume that the given two
matrices A1,⁢A2 are subjected to perturbations of the form A1→A1+δ11B11+δ12B12,⁢A2→A2+δ21B21+δ22B22,
where(3.56)B11=(12101),B12=(113112),B21=(011314),B22=(121311).Then(3.57)(I−A1−A2)−1B=(10227181116312).By Theorem 3.8, we get(3.58)rℂδ=rℝδ=r+δ=24283+81753.



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