# Stability of Matrix Polytopes with a Dominant   Vertex  and Implications for System Dynamics

Authors: Octavian Pastravanu; Mihaela-Hanako Matcovschi
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

The paper considers the class of matrix polytopes with a dominant vertex and the class of uncertain dynamical systems defined in discrete time and continuous time, respectively, by such polytopes. We analyze the standard concept of stability in the sense of Schur—abbreviated as SS  (resp., Hurwitz—abbreviated as HS), and we develop a general framework for the investigation of the diagonal stability relative to an arbitrary Hölderp-norm, 1≤p≤∞, abbreviated as SDSp (resp., HDSp). Our framework incorporates, as the particular case with p=2, the known condition of quadratic stability satisfied by a diagonal positive-definite matrix, i.e. SDS2 (resp., HDS2) means that the standard inequality of Stein (resp., Lyapunov) associated with all matrices of the polytope has a common diagonal solution. For the considered class of matrix polytopes, we prove the equivalence between SS and SDSp (resp., HS and HDSp), 1≤p≤∞ (fact which is not true for matrix polytopes with arbitrary structures). We show that the dominant vertex provides all the information needed for testing these stability properties and for computing the corresponding robustness indices. From the dynamical point of view, if an uncertain system is defined by a polytope with a dominant vertex, then the standard asymptotic stability ensures supplementary properties for the state-space trajectories, which refer to special types of Lyapunov functions and contractive invariant sets (characterized through vector p-norms weighted by diagonal positive-definite matrices). The applicability of the main results is illustrated by two numerical examples that cover both discrete- and continuous-time cases for the class of uncertain dynamics studied in our paper.


---

## Body

## 1. Introduction

### 1.1. Research Context and Objective

Consider the matrix polytope(1)𝒜={A∈ℝn×n∣A=∑k=1KγkAk,γk≥0,∑k=1Kγk=1},
where {A1,A2,…,AK} is a finite set of real n×n matrices. The Schur (resp., Hurwitz) stability—abbreviated as SS (resp., HS)—has been investigated for matrix polytope (1) starting with the 80s, by papers 4such as [1–13]. Research was strongly motivated by the dynamics analysis of linear systems with model uncertainties (which inherently occur due to incomplete or approximate information on process parameters). Description (1) is also referred to as a “polytopic matrix”, and from the modeling-power point of view, it incorporates the class of  “interval matrices,” defined by hyperrectangles in ℝn×n, i.e. 								(2)𝒜I={A∈ℝn×n∣A0-R≤A≤A0+R},A0,R∈ℝn×n,R≥0,
where the inequalities have a componentwise meaning. Significant results on Schur and Hurwitz  stability of interval matrices have been reported in [10, 14–24].Also starting with the 80s the linear algebra literature developed studies ona stronger type of matrix stability, called “diagonal stability”; pioneering works such as [25, 26] should be mentioned. In accordance with the monograph [27], a square matrix is Schur (resp., Hurwitz) diagonally stable if the Stein (resp., Lyapunov), inequality associated with that matrix has diagonal positive-definite solutions. As a natural expansion, our work [28] introduced the Stein (resp., Lyapunov), inequalities relative to a Hölderp-norm, 1≤p≤∞, and generalized the aforementioned diagonal stability concept to “Schur (resp., Hurwitz) diagonal stability relative to a Hölder  p-norm”—abbreviated as SDSp (resp., HDSp). For p=2, the framework proposed by [28] coincides with the classic approach presented by [27].TheSDSp (resp., HDSp), 1≤p≤∞, has been recently explored by our papers [29, 30] for interval matrices and for arbitrary polytopic matrices, respectively. It is worth saying that the monograph [27] addressed the standard case of diagonal stability (i.e. SDS2 and HDS2 in our nomenclature) for interval matrices.During the last decade, diagonal stability ensured a visible research potential for systems and control engineering, mainly related to the simpler form of the Lyapunov function candidates, as outlined by works such as [27–29, 31–35]. These works use the same terminology “diagonal stability” in the sense of a system property that is induced by the original matrix property discussed in the previous paragraphs.Our current paper focuses on the stability of a class of matrix polytopes of form (1) called “with a dominant vertex”—the concept is to be rigorously introduced by Definition 3 in the next section. We also study the dynamics of the polytopic systems associated with this class of matrix polytopes, described by the following equations:(i)
in the discrete-time case,(3-S)x(t+1)=Ax(t),A∈𝒜,x(t0)=x0,t,t0∈ℤ+,t≥t0,(ii)
in the continuous-time case,(3-H)x˙(t)=Ax(t),A∈𝒜,x(t0)=x0,t,t0∈ℝ+,t≥t0.In both models(3-S) and (3-H), the entries of matrix A are considered fixed (not time varying); they are uncertain in the sense that their values are incompletely known but surely satisfy the condition A∈𝒜. In other words, a single matrix A is used for modeling a certain evolution of the process, whereas for modeling two different evolutions (taking place separately) two distinct matrices A1≠A2, A1,A2∈𝒜 may be needed.


### 1.2. Paper Structure

For a matrix polytope (1) with a dominant vertex, we prove that SS (resp., HS) is equivalent to SDSp (resp., HDSp), 1≤p≤∞, unlike the case of an arbitrary matrix polytope (e.g., [30]) where (i) SDSp (resp., HDSp) is more conservative than SS (resp., HS) and (ii) results on SDSp1 and SDSp2 (resp., HDSp1 and HDSp2) may be different for p1≠p2, 1≤p1,  p2≤∞. These aspects are discussed by Section 2 of our work. Section 3 analyzes the implications of Section 2 for the dynamics of a polytopic system (3-S), respectively (3-H), defined by a matrix polytope with a dominant vertex. We show that the asymptotic stability of such a system is equivalent to the existence of Lyapunov functions and contractive invariant sets expressed in terms of any Hölder p-norm, by using an appropriate weighting matrix of diagonal form (whose positive entries depend on the chosen norm). The utility of our main results is illustrated in Section 4 by numerical examples, covering Schur (resp., Hurwitz) stability for matrix polytopes with a dominant vertex, as well as the implications for the dynamics of discrete-time (resp., continuous-time), polytopic systems.Throughout the text, in equation numbering we use the extension S (resp., H), for referring to Schur (resp., Hurwitz) stability and/or to discrete-time (resp., continuous-time), dynamics—as in the above equation(3-S) (resp., (3-H)). The extensions (S) and (H) play the same role for the labels of definitions and theorems.To ensure the fluent presentation of our results, their proofs are given in the Appendix.


### 1.3. Notations and Nomenclature

Letx=[x1⋯xn]T, y=[y1⋯yn]T∈ℝn be vectors.(i)
∥x∥p is the Hölder vector p-norm defined by ∥x∥p=[|x1|p+⋯+|xn|p]1/p for 1≤p<∞ and by ∥x∥∞=max1≤i≤n|xi| for p=∞.(ii)
“x≤y”, “x<y” mean componentwise inequalities, i.e.  xi≤yi, xi<yi, i=1,…,n.LetM=[mij], Q=[qij]∈ℝn×n be square matrices.(iii)
∥M∥p is the matrix norm induced by the vector p-norm through ∥M∥p=supx≠0(∥Mx∥p/∥x∥p)=max∥x∥p=1∥Mx∥p.(iv)
μp(M)=limh↓0h-1(∥I+hM∥p-1) is the matrix measure [36, page 41], based on the matrix norm ∥∥p.(v)
IfD=diag{d1,…,dn}, di>0, i=1,…,n, then the following regions of the complex plane Gjc(D-1MD)={z∈ℂ∣|z-mjj|≤∑i=1,i≠jn((dj/di)|mij|)}, j=1,…,n, are called the generalized Gershgorin's disks of M defined with D for columns.(vi)
IfD=diag{d1,…,dn}, di>0, i=1,…,n, then the following regions of the complex plane Gir(D-1MD)={z∈ℂ∣|z-mii|≤∑j=1,j≠in((dj/di)|mij|)}, i=1,…,n, are called the generalized Gershgorin’s disks of M defined with D for rows.(vii)
σ(M)={z∈ℂ∣det(zI-M)=0} is the spectrum of M, and λi(M)∈σ(M), i=1,…,n, are the eigenvalues of M.(viii)
Ifσ(M)⊂ℂS={z∈ℂ∣|z|<1}, then matrix M is said to be Schur stable (abbreviated as SS).(ix)
Ifσ(M)⊂ℂH={z∈ℂ∣Rez<0}, then matrix M is said to be Hurwitz stable (abbreviated as HS).(x)
IfM is nonnegative (all entries are nonnegative), its spectral radius is a positive eigenvalue, denoted by λmax(M), such that |λi(M)|≤λmax(M), i=1,…,n.(xi)
IfM is essentially nonnegative (all off-diagonal entries are nonnegative), then it has a real eigenvalue, denoted by λmax(M), such that Re{λi(M)}≤λmax(M), i=1,…,n—for example, Lemma 1 in [28].(xii)
IfM is symmetrical, then all its eigenvalues are real and there exists an eigenvalue denoted by λmax(M), such that λi(M)≤λmax(M), i=1,…,n.(xiii)
“M≻0", “M≺0" mean that M is a positive-definite, negative-definite matrix.(xiv)
If the oriented graph ofM is strongly connected, then M is called irreducible; otherwise M is called reducible.(xv)
Forp∈{1,2,∞}, the matrix norms ∥M∥p and matrix measures μp(M) have the following expressions:
(4-S)∥M∥p={max1≤j≤n{∑i=1n|mij|},p=1,λmax(MTM),p=2,max1≤i≤n{∑j=1n|mij|},p=∞,(4-H)μp(M)={max1≤j≤n{mjj+∑i=1,i≠jn|mij|},p=1,12λmax(M+MT),p=2,max1≤i≤n{mii+∑j=1,j≠in|mij|},p=∞.(xvi)
|M| denotes the matrix built with the absolute values of the entries of M.(xvii)
MS∈ℝn×n (S superscript from Schur) denotes the nonnegative matrix defined by MS=|M|.(xviii)
MH∈ℝn×n (H superscript from Hurwitz) denotes the essentially nonnegative matrix defined by MH=Md+|Mo|, where Md=diag{m11,...,mnn} and Mo=M-Md.(xix)
“M≤Q”, “M<Q” mean componentwise inequalities, i.e. mij≤qij, mij<qij, i,j=1,…,n.Throughout the text we shall write “X (resp., Y)” wherever “X” and “Y” are referred to in parallel.


## 1.1. Research Context and Objective

Consider the matrix polytope(1)𝒜={A∈ℝn×n∣A=∑k=1KγkAk,γk≥0,∑k=1Kγk=1},
where {A1,A2,…,AK} is a finite set of real n×n matrices. The Schur (resp., Hurwitz) stability—abbreviated as SS (resp., HS)—has been investigated for matrix polytope (1) starting with the 80s, by papers 4such as [1–13]. Research was strongly motivated by the dynamics analysis of linear systems with model uncertainties (which inherently occur due to incomplete or approximate information on process parameters). Description (1) is also referred to as a “polytopic matrix”, and from the modeling-power point of view, it incorporates the class of  “interval matrices,” defined by hyperrectangles in ℝn×n, i.e. 								(2)𝒜I={A∈ℝn×n∣A0-R≤A≤A0+R},A0,R∈ℝn×n,R≥0,
where the inequalities have a componentwise meaning. Significant results on Schur and Hurwitz  stability of interval matrices have been reported in [10, 14–24].Also starting with the 80s the linear algebra literature developed studies ona stronger type of matrix stability, called “diagonal stability”; pioneering works such as [25, 26] should be mentioned. In accordance with the monograph [27], a square matrix is Schur (resp., Hurwitz) diagonally stable if the Stein (resp., Lyapunov), inequality associated with that matrix has diagonal positive-definite solutions. As a natural expansion, our work [28] introduced the Stein (resp., Lyapunov), inequalities relative to a Hölderp-norm, 1≤p≤∞, and generalized the aforementioned diagonal stability concept to “Schur (resp., Hurwitz) diagonal stability relative to a Hölder  p-norm”—abbreviated as SDSp (resp., HDSp). For p=2, the framework proposed by [28] coincides with the classic approach presented by [27].TheSDSp (resp., HDSp), 1≤p≤∞, has been recently explored by our papers [29, 30] for interval matrices and for arbitrary polytopic matrices, respectively. It is worth saying that the monograph [27] addressed the standard case of diagonal stability (i.e. SDS2 and HDS2 in our nomenclature) for interval matrices.During the last decade, diagonal stability ensured a visible research potential for systems and control engineering, mainly related to the simpler form of the Lyapunov function candidates, as outlined by works such as [27–29, 31–35]. These works use the same terminology “diagonal stability” in the sense of a system property that is induced by the original matrix property discussed in the previous paragraphs.Our current paper focuses on the stability of a class of matrix polytopes of form (1) called “with a dominant vertex”—the concept is to be rigorously introduced by Definition 3 in the next section. We also study the dynamics of the polytopic systems associated with this class of matrix polytopes, described by the following equations:(i)
in the discrete-time case,(3-S)x(t+1)=Ax(t),A∈𝒜,x(t0)=x0,t,t0∈ℤ+,t≥t0,(ii)
in the continuous-time case,(3-H)x˙(t)=Ax(t),A∈𝒜,x(t0)=x0,t,t0∈ℝ+,t≥t0.In both models(3-S) and (3-H), the entries of matrix A are considered fixed (not time varying); they are uncertain in the sense that their values are incompletely known but surely satisfy the condition A∈𝒜. In other words, a single matrix A is used for modeling a certain evolution of the process, whereas for modeling two different evolutions (taking place separately) two distinct matrices A1≠A2, A1,A2∈𝒜 may be needed.


## 1.2. Paper Structure

For a matrix polytope (1) with a dominant vertex, we prove that SS (resp., HS) is equivalent to SDSp (resp., HDSp), 1≤p≤∞, unlike the case of an arbitrary matrix polytope (e.g., [30]) where (i) SDSp (resp., HDSp) is more conservative than SS (resp., HS) and (ii) results on SDSp1 and SDSp2 (resp., HDSp1 and HDSp2) may be different for p1≠p2, 1≤p1,  p2≤∞. These aspects are discussed by Section 2 of our work. Section 3 analyzes the implications of Section 2 for the dynamics of a polytopic system (3-S), respectively (3-H), defined by a matrix polytope with a dominant vertex. We show that the asymptotic stability of such a system is equivalent to the existence of Lyapunov functions and contractive invariant sets expressed in terms of any Hölder p-norm, by using an appropriate weighting matrix of diagonal form (whose positive entries depend on the chosen norm). The utility of our main results is illustrated in Section 4 by numerical examples, covering Schur (resp., Hurwitz) stability for matrix polytopes with a dominant vertex, as well as the implications for the dynamics of discrete-time (resp., continuous-time), polytopic systems.Throughout the text, in equation numbering we use the extension S (resp., H), for referring to Schur (resp., Hurwitz) stability and/or to discrete-time (resp., continuous-time), dynamics—as in the above equation(3-S) (resp., (3-H)). The extensions (S) and (H) play the same role for the labels of definitions and theorems.To ensure the fluent presentation of our results, their proofs are given in the Appendix.


## 1.3. Notations and Nomenclature

Letx=[x1⋯xn]T, y=[y1⋯yn]T∈ℝn be vectors.(i)
∥x∥p is the Hölder vector p-norm defined by ∥x∥p=[|x1|p+⋯+|xn|p]1/p for 1≤p<∞ and by ∥x∥∞=max1≤i≤n|xi| for p=∞.(ii)
“x≤y”, “x<y” mean componentwise inequalities, i.e.  xi≤yi, xi<yi, i=1,…,n.LetM=[mij], Q=[qij]∈ℝn×n be square matrices.(iii)
∥M∥p is the matrix norm induced by the vector p-norm through ∥M∥p=supx≠0(∥Mx∥p/∥x∥p)=max∥x∥p=1∥Mx∥p.(iv)
μp(M)=limh↓0h-1(∥I+hM∥p-1) is the matrix measure [36, page 41], based on the matrix norm ∥∥p.(v)
IfD=diag{d1,…,dn}, di>0, i=1,…,n, then the following regions of the complex plane Gjc(D-1MD)={z∈ℂ∣|z-mjj|≤∑i=1,i≠jn((dj/di)|mij|)}, j=1,…,n, are called the generalized Gershgorin's disks of M defined with D for columns.(vi)
IfD=diag{d1,…,dn}, di>0, i=1,…,n, then the following regions of the complex plane Gir(D-1MD)={z∈ℂ∣|z-mii|≤∑j=1,j≠in((dj/di)|mij|)}, i=1,…,n, are called the generalized Gershgorin’s disks of M defined with D for rows.(vii)
σ(M)={z∈ℂ∣det(zI-M)=0} is the spectrum of M, and λi(M)∈σ(M), i=1,…,n, are the eigenvalues of M.(viii)
Ifσ(M)⊂ℂS={z∈ℂ∣|z|<1}, then matrix M is said to be Schur stable (abbreviated as SS).(ix)
Ifσ(M)⊂ℂH={z∈ℂ∣Rez<0}, then matrix M is said to be Hurwitz stable (abbreviated as HS).(x)
IfM is nonnegative (all entries are nonnegative), its spectral radius is a positive eigenvalue, denoted by λmax(M), such that |λi(M)|≤λmax(M), i=1,…,n.(xi)
IfM is essentially nonnegative (all off-diagonal entries are nonnegative), then it has a real eigenvalue, denoted by λmax(M), such that Re{λi(M)}≤λmax(M), i=1,…,n—for example, Lemma 1 in [28].(xii)
IfM is symmetrical, then all its eigenvalues are real and there exists an eigenvalue denoted by λmax(M), such that λi(M)≤λmax(M), i=1,…,n.(xiii)
“M≻0", “M≺0" mean that M is a positive-definite, negative-definite matrix.(xiv)
If the oriented graph ofM is strongly connected, then M is called irreducible; otherwise M is called reducible.(xv)
Forp∈{1,2,∞}, the matrix norms ∥M∥p and matrix measures μp(M) have the following expressions:
(4-S)∥M∥p={max1≤j≤n{∑i=1n|mij|},p=1,λmax(MTM),p=2,max1≤i≤n{∑j=1n|mij|},p=∞,(4-H)μp(M)={max1≤j≤n{mjj+∑i=1,i≠jn|mij|},p=1,12λmax(M+MT),p=2,max1≤i≤n{mii+∑j=1,j≠in|mij|},p=∞.(xvi)
|M| denotes the matrix built with the absolute values of the entries of M.(xvii)
MS∈ℝn×n (S superscript from Schur) denotes the nonnegative matrix defined by MS=|M|.(xviii)
MH∈ℝn×n (H superscript from Hurwitz) denotes the essentially nonnegative matrix defined by MH=Md+|Mo|, where Md=diag{m11,...,mnn} and Mo=M-Md.(xix)
“M≤Q”, “M<Q” mean componentwise inequalities, i.e. mij≤qij, mij<qij, i,j=1,…,n.Throughout the text we shall write “X (resp., Y)” wherever “X” and “Y” are referred to in parallel.


## 2. Results on Matrix Polytopes

The current section explores the stability of matrix polytopes with a dominant vertex. For this class of polytopes, the standard Schur (Hurwitz) stability is proved to be equivalent to stronger stability properties, namely, diagonal stability relative to arbitrary Hölderp-norms 1≤p≤∞.


### 2.1. Preliminaries

Definition 1 (S).
Let us consider:1≤p≤∞; an arbitrary matrix A∈ℝn×n; a matrix polytope 𝒜  of  form (1); a nonsingular matrix Q∈ℝn×n.
(a) The inequality(5-S)∥Q-1AQ∥p<1
is called the Stein-type inequality relative to the p-norm associated with matrix A; matrix Q is said to be a solution to this inequality.
(b) MatrixQ is said to be a solution to the Stein-type inequality relative to the p-norm associated with the polytope 𝒜 if the following condition is fulfilled:
(6-S)∀A∈𝒜:∥Q-1AQ∥p<1.Definition 1 (H).
Let us consider:1≤p≤∞; an arbitrary matrix  A∈ℝn×n; a matrix polytope  𝒜 of form (1); a nonsingular matrix Q∈ℝn×n.
(a) The inequality(5-H)μp(Q-1AQ)<0
is called the Lyapunov-type inequality relative to the p-norm associated with matrix A; matrix Q is said to be a solution to this inequality.
(b) MatrixQ is said to be a solution to the Lyapunov-type inequality relative to the p-norm associated with the polytope 𝒜 if the following condition is fulfilled:
(6-H)∀A∈𝒜:μp(Q-1AQ)<0.Remark 1.
(i) The terminology introduced by Definition1(S)(a)  (resp., Definition 1(H)(a)) is motivated by the fact that inequality (5-S) (resp., (5-H)) with p=2 is equivalent to the standard Stein inequality
(7-S)ATPA-P≺0,P=(Q-1)T(Q-1),
respectively  the standard Lyapunov inequality(7-H)ATP+PA≺0,P=(Q-1)T(Q-1).
Indeed, for(5-S) with p=2 we may write that ∥Q-1AQ∥2<1 is equivalent to
(8-S)∀i=1,…,n:λi(QTAT(Q-1)TQ-1AQ)<1⟺λi(QTAT(Q-1)TQ-1AQ-I)<0⟺λi(AT(Q-1)T(Q-1)A-(Q-1)T(Q-1))<0,
which means the fulfillment of (7-S).
Similarly, for(5-H) with p=2 we may write that μ2(Q-1AQ)<0 is equivalent to
(8-H)∀i=1,…,n:λi(QTAT(Q-1)T+Q-1AQ)<0⟺λi(AT(Q-1)T(Q-1)+(Q-1)T(Q-1)A)<0,
which means the fulfillment of (7-H).
(ii) The existence ofP≻0 solving the standard Stein inequality (7-S) (resp., Lyapunov inequality (7-H) is equivalent to Schur (resp., Hurwitz) stability of matrix A.
(iii) Conditions(6-S) (resp., (6-H)) with p=2 in Definition 1(S)(b) (resp., Definition 1(H)(b)) represent the definition of Schur (resp., Hurwitz) quadratic stability of the matrix polytope 𝒜, for example, [37, page 213].Definition 2 (S).
Let𝒜 be a matrix polytope of form (1).
(a)𝒜 is called Schur stable (abbreviated as SS) if
(9-S)∀A∈𝒜:AisSS.
(b) Let1≤p≤∞. 𝒜 is called Schur diagonally stable relative to the p-norm (abbreviated as SDSp) if there exists a diagonal positive-definite matrix D≻0 that satisfies the Stein-type inequality relative to the p-norm associated with the polytope 𝒜, i.e. 									(10-S)∀A∈𝒜:∥D-1AD∥p<1.Definition 2 (H).
Let𝒜 be a matrix polytope of form (1).
(a)𝒜 is called Hurwitz stable (abbreviated as HS) if
(9-H)∀A∈𝒜:AisHS.
(b) Let1≤p≤∞. 𝒜 is called Hurwitz diagonally stable relative to the p-norm (abbreviated as HDSp) if there exists a diagonal positive-definite matrix D≻0 that satisfies the Lyapunov-type inequality relative to the p-norm associated with the polytope 𝒜, i.e. 									(10-H)∀A∈𝒜:μp(D-1AD)<0.Remark 2.
(i) If𝒜 is a trivial polytope defined by a single matrix A (i.e. A1=A, K=1, in (1)), then Definition 2(S)(b) (resp., Definition 2(H)(b)) coincides with Definition 1 in [28].
(ii) Let1≤p≤∞. If 𝒜 is a proper polytope (i.e. ∃k1≠k2, k1,k2∈{1,…,K}: Ak1≠Ak2 in (1)), then Definition 2(S)(b) (resp., Definition 2(H)(b)) proposes a meaningful extension of Definition 1 in [28]. Indeed, the simple use of Definition 1 in [28] does not necessarily imply the existence of a unique diagonal matrix D≻0 that satisfies inequality (10-S) (resp., (10-H)) for all matrices A∈𝒜.
(iii) Let1≤p≤∞. The SDSp  (resp., HDSp) is a property of 𝒜 stronger than SS (resp., HS). Indeed, each matrix A∈𝒜 is SS (resp., HS) once it is SDSp (resp., HDSp) in accordance with Remark 2 in [28].
(iv) Letp=2. Definition 2(S)(b) (resp., Definition 2(H)(b)) expresses a particular case of Schur (resp., Hurwitz) quadratic stability of the matrix polytope 𝒜—see Remark 1(iii). Subsequently, the quadratic stability is a property of 𝒜 stronger than SS (resp., HS) but, at the same time, weaker than SDS2 (resp., HDS2).Definition 3 (S).
Let𝒜 be a matrix polytope of form (1). If there exists a subscript k*∈{1,…,K}, such that the vertex Ak* fulfills one of the following two sets of componentwise inequalities:
(11-S-1)AkS≤Ak*,k≠k*,k=1,…,K,(11-S-2)AkS≤-Ak*,k≠k*,k=1,…,K,
then 𝒜 is called a matrix polytope with an S-dominant vertex, and Ak* is called the S-dominant vertex of 𝒜. (The notation MS is introduced in Section 1.3.)Definition 3 (H).
If there exists a subscriptk*∈{1,…,K}, such that the vertex Ak* fulfills the componentwise inequalities:
(11-H)AkH≤Ak*,k≠k*,k=1,…,K,
then 𝒜 is called a matrix polytope with an H-dominant vertex, and Ak* is called the H-dominant vertex of 𝒜. (The notation MH is introduced in Section 1.3.)Remark 3.
(i) If𝒜 is a matrix polytope with an S-dominant (resp., H-dominant) vertex Ak*, then Definition 3(S) (resp., Definition 3(H)) shows that matrix Ak* is nonnegative—inequalities (11-S-1) or nonpositive—inequalities (11-S-2) (resp., essentially nonnegative—inequalities (11-H)).
(ii) In the remainder of the text, we mainly address the case of the S-dominant vertex defined by inequalities(11-S-1). The case based on inequalities (11-S-2) does not require a separate approach, since all the results we are going to use for Ak* nonnegative remain valid for -Ak* nonnegative.
(iii) A matrix polytope𝒜 may have two S-dominant vertices denoted as Ak*,Ak** in the particular case when Ak*=-Ak**, Ak*≥0 satisfies inequalities (11-S-1), Ak**≤0 satisfies inequalities (11-S-2). We still can refer to 𝒜 as having “an S-dominant vertex,” since the stability properties of 𝒜 induced by Ak* and by Ak** are identical, as resulting from the further development of our paper.


### 2.2. Stability Analysis

Theorem 1 (S).
Let us consider:1≤p≤∞; a matrix polytope 𝒜 with an S-dominant vertex Ak*.
The following statements are equivalent.(i)
Ak* is SS.(ii)
𝒜 is SS.(iii)
There exists ap, 1≤p≤∞, such that 𝒜 is SDSp.(iv)
𝒜 is SDSp for all p, 1≤p≤∞.(v)
There exists a diagonal matrixD≻0 such that the union for all A∈𝒜 of the generalized Gershgorin’s disks written for columns is located inside the unit circle of the complex plane, i.e. ⋃A∈𝒜⋃j=1nGjc(D-1AD)⊂ℂS.(vi)
There exists a diagonal matrixD≻0 such that the union for all A∈𝒜 of the generalized Gershgorin’s disks written for rows is located inside the unit,  circle of the complex plane, i.e. ⋃A∈𝒜⋃i=1nGir(D-1AD)⊂ℂS.Proof.
See  the Appendix.Theorem 1 (H).
Let us consider:1≤p≤∞; a matrix polytope 𝒜 with an H-dominant vertex Ak*.
The following statements are equivalent.(i)
Ak* is HS.(ii)
𝒜 is HS.(iii)
There exists ap, 1≤p≤∞, such that 𝒜 is HDSp.(iv)
𝒜 is HDSp for all p, 1≤p≤∞.(v)
There exists a diagonal matrixD≻0 such that the union for all A∈𝒜 of the generalized Gershgorin’s disks written for columns is located in the left half plane of the complex plane, i.e. ⋃A∈𝒜⋃j=1nGjc(D-1AD)⊂ℂH.(vi)
There exists a diagonal matrixD≻0 such that the union for all A∈𝒜 of the generalized Gershgorin’s disks written for rows is located in the left half plane of the complex plane,  i.e. ⋃A∈𝒜⋃i=1nGir(D-1AD)⊂ℂH.Proof.
See the Appendix.


### 2.3. Diagonal Solutions to Stein-Type and Lyapunov-Type Inequalities

The S-dominant (resp., H-dominant) vertexAk* of a matrix polytope 𝒜 can be used not only for testing the properties SDSp (resp., HDSp) of 𝒜 but also for finding concrete diagonal matrices D≻0 that satisfy the inequality (10-S) in Definition 2(S) (resp., inequality  (10-H) in Definition 2(H)).Theorem 2 (S).
Let us consider:1≤p≤∞; a matrix polytope 𝒜 with an S-dominant vertex Ak*; a diagonal positive-definite matrix  D≻0.
MatrixD is a solution to (10-S) (i.e. D satisfies the Stein-type inequality relative to the p-norm associated with the polytope 𝒜) if and only if
(12-S)∥D-1Ak*D∥p<1
(i.e. D satisfies the Stein-type inequality relative to the p-norm associated with the S-dominant vertex Ak*).Proof.
See the Appendix.Theorem 2 (H).
Let us consider:1≤p≤∞; a matrix polytope 𝒜 with an H-dominant vertex Ak*;  a diagonal positive-definite matrix   D≻0.
MatrixD is a solution to (10-H) (i.e. D satisfies the Lyapunov-type inequality relative to the p-norm associated with the polytope 𝒜) if and only if
(12-H)μp(D-1Ak*D)<0
(i.e. D satisfies the Lyapunov-type inequality relative to the p-norm associated with the H-dominant vertex Ak*).Proof.
See the Appendix.Remark 4.
Let1≤p≤∞. Whenever Ak* is SS (resp., HS)  diagonal matrices D≻0 that satisfy (12-S)  (resp., (12-H)<?cmd?>) can be built along the lines of Lemma 3 and Remark 3 in [28]. Further comments are available in the next section that discloses the role of D≻0 in the dynamics of a polytopic system of form (3-S) (resp., (3-H)).


### 2.4. Stability Margins

The S-dominant (resp., H-dominant) vertexAk* of a matrix polytope 𝒜 also allows one to develop a robustness analysis for SS and SDSp of 𝒜 defined by (1) and (11-S-1) or (11-S-2)  (resp., HS and HDSp of 𝒜 defined by (1) and (11-H)).Definition 4 (S).
Let𝒜 be a matrix polytope with an S-dominant vertex Ak*.
(a) IfAk* is SS, then
(13-S)ρSS(Ak*)=1-maxi=1,…,n|λi(Ak*)|=1-λmax(Ak*)
is called the SS margin ofAk*.
(b) If𝒜 is SS, then
(14-S)ρSS(𝒜)=1-maxA∈𝒜maxi=1,…,n|λi(A)|
is called the SS margin of𝒜.
(c) Let1≤p≤∞. If 𝒜 is SDSp, then
(15-S)ρSDSp(𝒜)=1-infD≻0diagonalmaxA∈𝒜∥D-1AD∥p
is called the SDSp margin of 𝒜.Definition 4 (H).
Let𝒜 be a matrix polytope with an H-dominant vertex Ak*.
(a) IfAk* is HS, then
(13-H)ρHS(Ak*)=|maxi=1,…,nRe{λi(Ak*)}|=|λmax(Ak*)|
is called the HS margin ofAk*.
(b) If𝒜 is HS, then
(14-H)ρHS(𝒜)=|maxA∈𝒜maxi=1,…,nRe{λi(A)}|
is called the HS margin of𝒜.
(c) Let1≤p≤∞. If 𝒜 is HDSp, then
(15-H)ρHDSp(𝒜)=|infD≻0diagonalmaxA∈𝒜μp(D-1AD)|
is called the HDSp margin of 𝒜.Theorem 3 (S).
Let𝒜 be a matrix polytope with an S-dominant vertex Ak*. For any p, 1≤p≤∞, the following equalities hold:
(16-S)ρSDSp(𝒜)=ρSS(𝒜)=ρSS(Ak*).Proof.
See the Appendix.Theorem 3 (H).
Let𝒜 be a matrix polytope with an H-dominant vertex Ak*. For any p, 1≤p≤∞, the following equalities hold:
(16-H)ρHDSp(𝒜)=ρHS(𝒜)=ρHS(Ak*).Proof.
See the Appendix.Remark 5.
(i) For each stability property of the polytope𝒜 discussed in Section 2.2, the corresponding margin (also called “degree” in the control-engineering literature) quantifies the distance between a matrix A∈𝒜 representing the “worst case” relative to that property and the “limit situation” where that property is generically lost for an arbitrary matrix. Theorem 3(S)  (resp., Theorem 3(H)) shows that the “worst case” of 𝒜 relative to SS and SDSp (resp., HS and HDSp) is defined by the S-dominant  (resp., H-dominant) vertex.
(ii) Forp=1, Theorem 3(S)  (resp., Theorem 3(H)) ensure the existence of a diagonal matrix D≻0 such that the union for all A∈𝒜 of the generalized Gershgorin’s disks written for columns ⋃A∈𝒜⋃j=1nGjc(D-1AD) is located in the region of the complex plane defined by |z|≤1-ρSS(Ak*) (resp., Rez≤-ρSS(Ak*)). The same location also corresponds to the union for all A∈𝒜 of the generalized Gershgorin’s disks written for rows ⋃A∈𝒜⋃i=1nGir(D-1AD), where the existence of the diagonal matrix D≻0 is guaranteed by Theorem 3(S) (resp., Theorem 3(H)) with p=∞. Obviously, the region |z|≤1-ρSS(Ak*)  (resp., Rez≤-ρSS(Ak*)) refines the condition formulated by Theorem 1(S)(v)-(vi)  (resp., Theorem 1(H)(v)-(vi)) for the location of the generalized Gershgorin’s disks.
(iii) The equality(16-S) (resp., (16-H)) plays an important role in the characterization of the dynamic properties exhibited by the polytopic system (3-S)  (resp., (3-H)). Further details on this role are available in Remark 6 of the next section.
(iv) For an arbitrary polytope𝒜 (without a dominant vertex), equality (16-S) (resp., (16-H)) does not hold true, in general. If, for a given p, 1≤p≤∞, 𝒜 is SDSp (resp., HDSp) then ρSDSp(𝒜)≤ρSS(𝒜) (resp., ρHDSp(𝒜)≤ρHS(𝒜)), fact which was anticipated by Remark 2(iii) in general terms, without using this specific language of  “stability margins.” Moreover, if for given p1≠p2, 1≤p1, p2≤∞, 𝒜 is diagonally stable relative to both p1- and p2-norm, then we may have ρSDSp1(𝒜)≠ρSDSp2(𝒜)  (resp., ρHDSp1(𝒜)≠ρHDSp2(𝒜)), as already suggested by our recent paper [30].


### 2.5. Particular Case of Interval Matrices with a Dominant Vertex

Theorems1(S), 2(S), and 3(S) generalize the results reported in [27, Lemma 3.4.18], [29] for SS and SDSp of interval matrices of form (2) with A0 or -A0 nonnegative, because these two types of interval matrices represent particular cases of matrix polytopes with an S-dominant vertex defined by inequalities (11-S-1) or (11-S-2).Similarly, Theorems1(H), 2(H), and 3(H) generalize results reported in [29] for HS and HDSp of interval matrices of form (2) with A0 essentially nonnegative, since such interval matrices represent a particular case of matrix polytopes with an H-dominant vertex defined by inequalities (11-H).


## 2.1. Preliminaries

Definition 1 (S).
Let us consider:1≤p≤∞; an arbitrary matrix A∈ℝn×n; a matrix polytope 𝒜  of  form (1); a nonsingular matrix Q∈ℝn×n.
(a) The inequality(5-S)∥Q-1AQ∥p<1
is called the Stein-type inequality relative to the p-norm associated with matrix A; matrix Q is said to be a solution to this inequality.
(b) MatrixQ is said to be a solution to the Stein-type inequality relative to the p-norm associated with the polytope 𝒜 if the following condition is fulfilled:
(6-S)∀A∈𝒜:∥Q-1AQ∥p<1.Definition 1 (H).
Let us consider:1≤p≤∞; an arbitrary matrix  A∈ℝn×n; a matrix polytope  𝒜 of form (1); a nonsingular matrix Q∈ℝn×n.
(a) The inequality(5-H)μp(Q-1AQ)<0
is called the Lyapunov-type inequality relative to the p-norm associated with matrix A; matrix Q is said to be a solution to this inequality.
(b) MatrixQ is said to be a solution to the Lyapunov-type inequality relative to the p-norm associated with the polytope 𝒜 if the following condition is fulfilled:
(6-H)∀A∈𝒜:μp(Q-1AQ)<0.Remark 1.
(i) The terminology introduced by Definition1(S)(a)  (resp., Definition 1(H)(a)) is motivated by the fact that inequality (5-S) (resp., (5-H)) with p=2 is equivalent to the standard Stein inequality
(7-S)ATPA-P≺0,P=(Q-1)T(Q-1),
respectively  the standard Lyapunov inequality(7-H)ATP+PA≺0,P=(Q-1)T(Q-1).
Indeed, for(5-S) with p=2 we may write that ∥Q-1AQ∥2<1 is equivalent to
(8-S)∀i=1,…,n:λi(QTAT(Q-1)TQ-1AQ)<1⟺λi(QTAT(Q-1)TQ-1AQ-I)<0⟺λi(AT(Q-1)T(Q-1)A-(Q-1)T(Q-1))<0,
which means the fulfillment of (7-S).
Similarly, for(5-H) with p=2 we may write that μ2(Q-1AQ)<0 is equivalent to
(8-H)∀i=1,…,n:λi(QTAT(Q-1)T+Q-1AQ)<0⟺λi(AT(Q-1)T(Q-1)+(Q-1)T(Q-1)A)<0,
which means the fulfillment of (7-H).
(ii) The existence ofP≻0 solving the standard Stein inequality (7-S) (resp., Lyapunov inequality (7-H) is equivalent to Schur (resp., Hurwitz) stability of matrix A.
(iii) Conditions(6-S) (resp., (6-H)) with p=2 in Definition 1(S)(b) (resp., Definition 1(H)(b)) represent the definition of Schur (resp., Hurwitz) quadratic stability of the matrix polytope 𝒜, for example, [37, page 213].Definition 2 (S).
Let𝒜 be a matrix polytope of form (1).
(a)𝒜 is called Schur stable (abbreviated as SS) if
(9-S)∀A∈𝒜:AisSS.
(b) Let1≤p≤∞. 𝒜 is called Schur diagonally stable relative to the p-norm (abbreviated as SDSp) if there exists a diagonal positive-definite matrix D≻0 that satisfies the Stein-type inequality relative to the p-norm associated with the polytope 𝒜, i.e. 									(10-S)∀A∈𝒜:∥D-1AD∥p<1.Definition 2 (H).
Let𝒜 be a matrix polytope of form (1).
(a)𝒜 is called Hurwitz stable (abbreviated as HS) if
(9-H)∀A∈𝒜:AisHS.
(b) Let1≤p≤∞. 𝒜 is called Hurwitz diagonally stable relative to the p-norm (abbreviated as HDSp) if there exists a diagonal positive-definite matrix D≻0 that satisfies the Lyapunov-type inequality relative to the p-norm associated with the polytope 𝒜, i.e. 									(10-H)∀A∈𝒜:μp(D-1AD)<0.Remark 2.
(i) If𝒜 is a trivial polytope defined by a single matrix A (i.e. A1=A, K=1, in (1)), then Definition 2(S)(b) (resp., Definition 2(H)(b)) coincides with Definition 1 in [28].
(ii) Let1≤p≤∞. If 𝒜 is a proper polytope (i.e. ∃k1≠k2, k1,k2∈{1,…,K}: Ak1≠Ak2 in (1)), then Definition 2(S)(b) (resp., Definition 2(H)(b)) proposes a meaningful extension of Definition 1 in [28]. Indeed, the simple use of Definition 1 in [28] does not necessarily imply the existence of a unique diagonal matrix D≻0 that satisfies inequality (10-S) (resp., (10-H)) for all matrices A∈𝒜.
(iii) Let1≤p≤∞. The SDSp  (resp., HDSp) is a property of 𝒜 stronger than SS (resp., HS). Indeed, each matrix A∈𝒜 is SS (resp., HS) once it is SDSp (resp., HDSp) in accordance with Remark 2 in [28].
(iv) Letp=2. Definition 2(S)(b) (resp., Definition 2(H)(b)) expresses a particular case of Schur (resp., Hurwitz) quadratic stability of the matrix polytope 𝒜—see Remark 1(iii). Subsequently, the quadratic stability is a property of 𝒜 stronger than SS (resp., HS) but, at the same time, weaker than SDS2 (resp., HDS2).Definition 3 (S).
Let𝒜 be a matrix polytope of form (1). If there exists a subscript k*∈{1,…,K}, such that the vertex Ak* fulfills one of the following two sets of componentwise inequalities:
(11-S-1)AkS≤Ak*,k≠k*,k=1,…,K,(11-S-2)AkS≤-Ak*,k≠k*,k=1,…,K,
then 𝒜 is called a matrix polytope with an S-dominant vertex, and Ak* is called the S-dominant vertex of 𝒜. (The notation MS is introduced in Section 1.3.)Definition 3 (H).
If there exists a subscriptk*∈{1,…,K}, such that the vertex Ak* fulfills the componentwise inequalities:
(11-H)AkH≤Ak*,k≠k*,k=1,…,K,
then 𝒜 is called a matrix polytope with an H-dominant vertex, and Ak* is called the H-dominant vertex of 𝒜. (The notation MH is introduced in Section 1.3.)Remark 3.
(i) If𝒜 is a matrix polytope with an S-dominant (resp., H-dominant) vertex Ak*, then Definition 3(S) (resp., Definition 3(H)) shows that matrix Ak* is nonnegative—inequalities (11-S-1) or nonpositive—inequalities (11-S-2) (resp., essentially nonnegative—inequalities (11-H)).
(ii) In the remainder of the text, we mainly address the case of the S-dominant vertex defined by inequalities(11-S-1). The case based on inequalities (11-S-2) does not require a separate approach, since all the results we are going to use for Ak* nonnegative remain valid for -Ak* nonnegative.
(iii) A matrix polytope𝒜 may have two S-dominant vertices denoted as Ak*,Ak** in the particular case when Ak*=-Ak**, Ak*≥0 satisfies inequalities (11-S-1), Ak**≤0 satisfies inequalities (11-S-2). We still can refer to 𝒜 as having “an S-dominant vertex,” since the stability properties of 𝒜 induced by Ak* and by Ak** are identical, as resulting from the further development of our paper.


## 2.2. Stability Analysis

Theorem 1 (S).
Let us consider:1≤p≤∞; a matrix polytope 𝒜 with an S-dominant vertex Ak*.
The following statements are equivalent.(i)
Ak* is SS.(ii)
𝒜 is SS.(iii)
There exists ap, 1≤p≤∞, such that 𝒜 is SDSp.(iv)
𝒜 is SDSp for all p, 1≤p≤∞.(v)
There exists a diagonal matrixD≻0 such that the union for all A∈𝒜 of the generalized Gershgorin’s disks written for columns is located inside the unit circle of the complex plane, i.e. ⋃A∈𝒜⋃j=1nGjc(D-1AD)⊂ℂS.(vi)
There exists a diagonal matrixD≻0 such that the union for all A∈𝒜 of the generalized Gershgorin’s disks written for rows is located inside the unit,  circle of the complex plane, i.e. ⋃A∈𝒜⋃i=1nGir(D-1AD)⊂ℂS.Proof.
See  the Appendix.Theorem 1 (H).
Let us consider:1≤p≤∞; a matrix polytope 𝒜 with an H-dominant vertex Ak*.
The following statements are equivalent.(i)
Ak* is HS.(ii)
𝒜 is HS.(iii)
There exists ap, 1≤p≤∞, such that 𝒜 is HDSp.(iv)
𝒜 is HDSp for all p, 1≤p≤∞.(v)
There exists a diagonal matrixD≻0 such that the union for all A∈𝒜 of the generalized Gershgorin’s disks written for columns is located in the left half plane of the complex plane, i.e. ⋃A∈𝒜⋃j=1nGjc(D-1AD)⊂ℂH.(vi)
There exists a diagonal matrixD≻0 such that the union for all A∈𝒜 of the generalized Gershgorin’s disks written for rows is located in the left half plane of the complex plane,  i.e. ⋃A∈𝒜⋃i=1nGir(D-1AD)⊂ℂH.Proof.
See the Appendix.


## 2.3. Diagonal Solutions to Stein-Type and Lyapunov-Type Inequalities

The S-dominant (resp., H-dominant) vertexAk* of a matrix polytope 𝒜 can be used not only for testing the properties SDSp (resp., HDSp) of 𝒜 but also for finding concrete diagonal matrices D≻0 that satisfy the inequality (10-S) in Definition 2(S) (resp., inequality  (10-H) in Definition 2(H)).Theorem 2 (S).
Let us consider:1≤p≤∞; a matrix polytope 𝒜 with an S-dominant vertex Ak*; a diagonal positive-definite matrix  D≻0.
MatrixD is a solution to (10-S) (i.e. D satisfies the Stein-type inequality relative to the p-norm associated with the polytope 𝒜) if and only if
(12-S)∥D-1Ak*D∥p<1
(i.e. D satisfies the Stein-type inequality relative to the p-norm associated with the S-dominant vertex Ak*).Proof.
See the Appendix.Theorem 2 (H).
Let us consider:1≤p≤∞; a matrix polytope 𝒜 with an H-dominant vertex Ak*;  a diagonal positive-definite matrix   D≻0.
MatrixD is a solution to (10-H) (i.e. D satisfies the Lyapunov-type inequality relative to the p-norm associated with the polytope 𝒜) if and only if
(12-H)μp(D-1Ak*D)<0
(i.e. D satisfies the Lyapunov-type inequality relative to the p-norm associated with the H-dominant vertex Ak*).Proof.
See the Appendix.Remark 4.
Let1≤p≤∞. Whenever Ak* is SS (resp., HS)  diagonal matrices D≻0 that satisfy (12-S)  (resp., (12-H)<?cmd?>) can be built along the lines of Lemma 3 and Remark 3 in [28]. Further comments are available in the next section that discloses the role of D≻0 in the dynamics of a polytopic system of form (3-S) (resp., (3-H)).


## 2.4. Stability Margins

The S-dominant (resp., H-dominant) vertexAk* of a matrix polytope 𝒜 also allows one to develop a robustness analysis for SS and SDSp of 𝒜 defined by (1) and (11-S-1) or (11-S-2)  (resp., HS and HDSp of 𝒜 defined by (1) and (11-H)).Definition 4 (S).
Let𝒜 be a matrix polytope with an S-dominant vertex Ak*.
(a) IfAk* is SS, then
(13-S)ρSS(Ak*)=1-maxi=1,…,n|λi(Ak*)|=1-λmax(Ak*)
is called the SS margin ofAk*.
(b) If𝒜 is SS, then
(14-S)ρSS(𝒜)=1-maxA∈𝒜maxi=1,…,n|λi(A)|
is called the SS margin of𝒜.
(c) Let1≤p≤∞. If 𝒜 is SDSp, then
(15-S)ρSDSp(𝒜)=1-infD≻0diagonalmaxA∈𝒜∥D-1AD∥p
is called the SDSp margin of 𝒜.Definition 4 (H).
Let𝒜 be a matrix polytope with an H-dominant vertex Ak*.
(a) IfAk* is HS, then
(13-H)ρHS(Ak*)=|maxi=1,…,nRe{λi(Ak*)}|=|λmax(Ak*)|
is called the HS margin ofAk*.
(b) If𝒜 is HS, then
(14-H)ρHS(𝒜)=|maxA∈𝒜maxi=1,…,nRe{λi(A)}|
is called the HS margin of𝒜.
(c) Let1≤p≤∞. If 𝒜 is HDSp, then
(15-H)ρHDSp(𝒜)=|infD≻0diagonalmaxA∈𝒜μp(D-1AD)|
is called the HDSp margin of 𝒜.Theorem 3 (S).
Let𝒜 be a matrix polytope with an S-dominant vertex Ak*. For any p, 1≤p≤∞, the following equalities hold:
(16-S)ρSDSp(𝒜)=ρSS(𝒜)=ρSS(Ak*).Proof.
See the Appendix.Theorem 3 (H).
Let𝒜 be a matrix polytope with an H-dominant vertex Ak*. For any p, 1≤p≤∞, the following equalities hold:
(16-H)ρHDSp(𝒜)=ρHS(𝒜)=ρHS(Ak*).Proof.
See the Appendix.Remark 5.
(i) For each stability property of the polytope𝒜 discussed in Section 2.2, the corresponding margin (also called “degree” in the control-engineering literature) quantifies the distance between a matrix A∈𝒜 representing the “worst case” relative to that property and the “limit situation” where that property is generically lost for an arbitrary matrix. Theorem 3(S)  (resp., Theorem 3(H)) shows that the “worst case” of 𝒜 relative to SS and SDSp (resp., HS and HDSp) is defined by the S-dominant  (resp., H-dominant) vertex.
(ii) Forp=1, Theorem 3(S)  (resp., Theorem 3(H)) ensure the existence of a diagonal matrix D≻0 such that the union for all A∈𝒜 of the generalized Gershgorin’s disks written for columns ⋃A∈𝒜⋃j=1nGjc(D-1AD) is located in the region of the complex plane defined by |z|≤1-ρSS(Ak*) (resp., Rez≤-ρSS(Ak*)). The same location also corresponds to the union for all A∈𝒜 of the generalized Gershgorin’s disks written for rows ⋃A∈𝒜⋃i=1nGir(D-1AD), where the existence of the diagonal matrix D≻0 is guaranteed by Theorem 3(S) (resp., Theorem 3(H)) with p=∞. Obviously, the region |z|≤1-ρSS(Ak*)  (resp., Rez≤-ρSS(Ak*)) refines the condition formulated by Theorem 1(S)(v)-(vi)  (resp., Theorem 1(H)(v)-(vi)) for the location of the generalized Gershgorin’s disks.
(iii) The equality(16-S) (resp., (16-H)) plays an important role in the characterization of the dynamic properties exhibited by the polytopic system (3-S)  (resp., (3-H)). Further details on this role are available in Remark 6 of the next section.
(iv) For an arbitrary polytope𝒜 (without a dominant vertex), equality (16-S) (resp., (16-H)) does not hold true, in general. If, for a given p, 1≤p≤∞, 𝒜 is SDSp (resp., HDSp) then ρSDSp(𝒜)≤ρSS(𝒜) (resp., ρHDSp(𝒜)≤ρHS(𝒜)), fact which was anticipated by Remark 2(iii) in general terms, without using this specific language of  “stability margins.” Moreover, if for given p1≠p2, 1≤p1, p2≤∞, 𝒜 is diagonally stable relative to both p1- and p2-norm, then we may have ρSDSp1(𝒜)≠ρSDSp2(𝒜)  (resp., ρHDSp1(𝒜)≠ρHDSp2(𝒜)), as already suggested by our recent paper [30].


## 2.5. Particular Case of Interval Matrices with a Dominant Vertex

Theorems1(S), 2(S), and 3(S) generalize the results reported in [27, Lemma 3.4.18], [29] for SS and SDSp of interval matrices of form (2) with A0 or -A0 nonnegative, because these two types of interval matrices represent particular cases of matrix polytopes with an S-dominant vertex defined by inequalities (11-S-1) or (11-S-2).Similarly, Theorems1(H), 2(H), and 3(H) generalize results reported in [29] for HS and HDSp of interval matrices of form (2) with A0 essentially nonnegative, since such interval matrices represent a particular case of matrix polytopes with an H-dominant vertex defined by inequalities (11-H).


## 3. Results on Polytopic Systems

The current section shows that a polytopic system defined by a matrix polytope with a dominant vertex may exhibit dynamical properties stronger than the standard concept of asymptotic stability; these dynamical properties are correlated, by equivalence, to the algebraic properties of the dominant vertex.Theorem 4 (S).
Let us consider:1≤p≤∞;  a discrete-time polytopic system of form (3-S) where polytope 𝒜 has an S-dominant vertex Ak*; a positive-definite diagonal matrix    D≻0; a constant   r satisfying 0<r<1.
The following statements are equivalent:(i)
(17-S)∥D-1Ak*D∥p≤r.(ii)
For the polytopic system(3-S), the functions
(18-S)𝒱pα:ℝn⟶ℝ+,𝒱pα(x)=1α∥D-1x∥p,α>0are strong diagonal Lyapunov functions, with the decreasing rater, i.e. 										(19-S)∀x(t)isatrajectoryofsystem(3-S),∀t∈ℤ+,t≥t0:𝒱pα(x(t+1))≤r𝒱pα(x(t)).(iii)
The contractive sets(20-S)𝒳pα(t;t0)={x∈ℝn∣∥D-1x∥p≤αr(t-t0)},t,t0∈ℤ+,t≥t0,α>0are invariant with respect to the state-space trajectories (solutions) of the polytopic system(3-S), i.e. 										(21-S)∀α>0,∀t,t0∈ℤ+,t≥t0,∀x0=x(t0)∈ℝn:∥D-1x0∥p≤α⟹∥D-1x(t;t0,x0)∥p≤αr(t-t0).Proof.
See the Appendix.Theorem 4 (H).
Let us consider:1≤p≤∞; a continuous-time polytopic system of form (3-H) where polytope 𝒜 has an H-dominant vertex Ak*; a positive-definite diagonal matrix   D≻0; a constant   r satisfying r<0.
The following statements are equivalent:(i)
(17-H)μp(D-1Ak*D)≤r.(ii)
For the polytopic system(3-H), the functions
(18-H)𝒱pα:ℝn⟶ℝ+,𝒱pα(x)=1α∥D-1x∥p,α>0are strong diagonal Lyapunov functions, with the decreasing rater, i.e. 										(19-H)∀x(t)isatrajectoryofsystem(3-H),∀t∈ℝ+,t≥t0:D+𝒱pα(x(t))=limh↓01h[𝒱pα(x(t+h))-𝒱pα(x(t))]D+𝒱pα(x(t))≤r𝒱pα(x(t)).(iii)
The contractive sets(20-H)𝒳pα(t;t0)={x∈ℝn∣∥D-1x∥p≤αer(t-t0)},t,t0∈ℝ+,t≥t0,α>0are invariant with respect to the state-space trajectories (solutions) of the polytopic system(3-H), i.e. 										(21-H)∀α>0,∀t,t0∈ℝ+,t≥t0,∀x0=x(t0)∈ℝn:∥D-1x0∥p≤α⟹∥D-1x(t;t0,x0)∥≤αer(t-t0).Proof.
See the Appendix.Remark 6.
(i) Let1≤p≤∞. The exploration of the dynamical properties of a polytopic system via Theorem 4(S)  (resp., Theorem 4(H)) outlines the importance of the concrete value 0<r<1  (resp., r<0) in the right hand side of inequality (17-S)  (resp., (17-H)). This concrete value r does not appear explicitly in the Stein-type inequality (12-S)  (resp., Lyapunov-type inequality (12-H)); the existence of a diagonal matrix D≻0 that solves inequality (12-S)  (resp., (12-H)) represents a necessary and a sufficient condition for the SDSp (resp., HDSp) of a matrix polytope 𝒜. For a polytopic system (3-S) (resp., (3-H)) a complete description of the dynamics implies the knowledge of pairs formed by r and D that satisfy Theorem 4(S) (resp., Theorem 4(H)).
(ii) The constant0<r<1 (resp., r<0) in Theorem 4(S)  (resp., Theorem 4(H)) represents a decreasing rate for the diagonal Lyapunov functions and for the contractive invariant sets. We are going to prove that for any p, 1≤p≤∞, the value of the fastest decreasing rate is given by the λmax(Ak*), regardless of the discrete-time or continuous-time nature of the dynamics. Indeed, for r<λmax(Ak*), there exists no diagonal matrix D≻0 satisfying inequality (17-S) (resp., (17-H)). For r≥λmax(Ak*), we use Lemma 3 and Remark 3 in [28] that yield the following discussion on the irreducibility/reducibility of Ak*.Case 1 (matrixAk* is irreducible).
(The definition of an irreducible matrix is introduced in Section1.3.) Denote by v=[v1⋯vn]T>0 and w=[w1⋯wn]T>0 its right and left Perron eigenvectors, respectively. Given p, 1≤p≤∞, we construct the diagonal matrix Dp=diag{v11/q/w11/p,…,vn1/q/wn1/p}≻0, where (i) (1/p)+(1/q)=1 if 1<p<∞; (ii) 1/p=1, 1/q=0 if p=1; (iii) 1/p=0, 1/q=1 if p=∞. Matrix Dp≻0 fulfills the following equality:(i)
∥Dp-1Ak*Dp∥p=λmax(Ak*) if Ak* is nonnegative, i.e. Ak* is defined by (11-S-1);(ii)
∥Dp-1Ak*Dp∥p=λmax(-Ak*) if Ak* is nonpositive, i.e. Ak* is defined by (11-S-2);(iii)
μp(Dp-1Ak*Dp)=λmax(Ak*) if Ak* is essentially nonnegative, i.e. Ak* is defined by (11-H).Case 2 (matrixAk* is reducible).
For anyr>λmax(Ak*), there exists ε>0 such that λmax(Ak*)<λmax(Ak*+εJ)<r, where J is the n by n matrix with all its entries 1. We apply the procedure presented by Case 1 to the irreducible matrix Ak*+εJ, and the resulting diagonal matrix Dp≻0 fulfills ∥Dp-1(Ak*+εJ)Dp∥p=λmax(Ak*+εJ)<r, μp(Dp-1(Ak*+εJ)Dp)=λmax(Ak*+εJ)<r, respectively. On the other hand, we have the norm inequality ∥Dp-1Ak*Dp∥p≤∥Dp-1(Ak*+εJ)Dp∥p, the measure inequality μp(Dp-1Ak*Dp)≤μp(Dp-1(Ak*+εJ)Dp), respectively. Subsequently, we obtain λmax(Ak*)<∥Dp-1Ak*Dp∥p<r, λmax(Ak*)<μp(Dp-1Ak*Dp)<r, respectively.
Thus, for anyp, 1≤p≤∞, we can construct a diagonal matrix Dp≻0 such that inequality (17-S)  (resp., (17-H)) is fulfilled with r=λmax(Ak*)—for Ak* irreducible and with r>λmax(Ak*) but as close to λmax(Ak*) as we want—for Ak* reducible.
(iii) The fastest decreasing rate of the diagonal Lyapunov functions and of the contractive invariant sets can be expressed in terms of the stability margins (discussed in Section2.4), as 1-ρSS(Ak*) for the discrete-time case and -ρSS(Ak*) for the continuous-time case. This point of view shows that the stability margins provide an algebraic characterization for the polytope 𝒜 and, concomitantly, allow the evaluation of the dynamical properties of the polytopic system (3-S)  (resp., (3-H)).
(iv) For an arbitrary polytope𝒜 (without a dominant vertex), the fastest decreasing rate may depend on the p-norm that defines the Lyapunov function and the invariant sets (if exist). If, for a given p,1≤p≤∞, 𝒜 is SDSp  (resp., HDSp) then the fastest decreasing rate corresponding to the p-norm of the polytopic system can be expressed in terms of the stability margins as 1-ρSDSp(𝒜) (resp., -ρHDSp(𝒜)). The fastest decreasing rate corresponding to the p-norm can be fairly estimated by a computational procedure based on a bisection method presented in [30].


## 4. Illustrative Examples

This section presents two examples that illustrate the usefulness of the theoretical results developed by our work. Example1 explores (i) the stability of a matrix polytope with an S-dominant vertex of form (1) and   (11-S-1); (ii) the dynamical properties of the discrete-time polytopic system of form (3-S) defined by the considered polytope. Example 2 explores (i) the stability of a matrix polytope with an H-dominant vertex of form (1) and (11-H); (ii) the dynamical properties of the continuous-time polytopic system of form (3-H) defined by the considered polytope.Example 1.
It is adapted from [27]. Let
(22)𝒮[-1,1]={S∈ℝn×n∣S=diag{s1,…,sn},|si|≤1,i=1,…,n}={S∈ℝn×n∣|S|≤In}
be the set of diagonal matrices whose elements are subunitary. 𝒮[-1,1] is the convex hull generated by the set of 2n vertices 𝒮sgn={S∈ℝn×n∣|S|=In}, also called the class of signature matrices.
Given a nonnegative matrixA*∈ℝn×n, the set
(23)𝒜=ℛ(A*)={A=A*S∣S∈𝒮[-1,1]}
is a matrix polytope of form (1) with the vertices Ai=A*Si, where Si∈𝒮sgn, i=1,…,2n. The set  𝒜=ℛ(A*) defined by (23) is a matrix polytope with an S-dominant vertex because matrix A* is a vertex that satisfies condition (11-S-1).
Theorem1(S) shows that the Schur stability of A* is a necessary and a sufficient condition for the Schur diagonal stability of the polytope 𝒜=ℛ(A*) relative to any p-norm, 1≤p≤∞. According to Theorem 2(S), a diagonal positive-definite matrix Dp≻0 satisfies the Stein-type inequality relative to the p-norm associated with 𝒜=ℛ(A*) if and only if Dp≻0 satisfies the Stein-type inequality relative to the p-norm associated with A*. This property of 𝒜=ℛ(A*) is guaranteed for any p-norm by Theorem 2(S), whereas Proposition 2.5.8 in [27] can guarantee only the particular case corresponding to p=2.
IfA* is Schur stable, then the eigenvalue λmax(A*) allows one to investigate the following properties: (i) for the polytope 𝒜=ℛ(A*), the SDSp margins are given by relation (16-S) in Theorem 3(S) i.e. ρSDSp(ℛ(A*))=1-λmax(A*), for any p, 1≤p≤∞; (ii) for the discrete-time polytopic system of form (3-S) defined by 𝒜=ℛ(A*), regardless of the p-norm considered in Theorem 4(S) for the Lyapunov function (18-S), and for the contractive invariant sets (20-S), the fastest decreasing rate is λmax(A*) if A* is irreducible and arbitrarily close to λmax(A*) if A* is reducible—as per Remark 6(ii).
Finally, we notice that the nonpositive matrixA**=-A* is also a vertex of the polytope 𝒜=ℛ(A*) and it satisfies the dominance condition (11-S-2). This means that the polytope 𝒜=ℛ(A*) fits in the particular context commented by Remark 3(iii). It is obvious that the analysis presented by the current example is complete, in the sense that the vertex A**≤0 brings no supplementary information (since the matrix -A** is nonnegative, there exists λmax(-A**)=λmax(A*), and for all p, 1≤p≤∞, the equality ∥D-1A**D∥p=∥D-1A*D∥p holds true).
The above approach appliesmutatis-mutandis to the investigation of the matrix polytope ℒ(A*)={A=SA*∣S∈𝒮[-1,1]}. In this case, Theorem 2(S) generalizes for 1≤p≤∞ the result that can be obtained when p=2 by using Proposition 2.5.9 in [27] for matrix A* and polytope ℒ(A*).Example 2.
Let us consider the interval matrix [14]:
(24)𝒜={A∈ℝ2×2∣A=A0+Δ,|Δ|≤R},A0=[-3.81.60.6-4.2],R=0.17[1111],
that is a matrix polytope with K=24 vertices:
(25)Ak=A0+Δk,k=1,…,16,whereΔk=0.17[±1±1±1±1].𝒜 has an H-dominant vertex
(26)A*=A0+R=[-3.631.770.77-4.03],
which satisfies inequalities (11-H). Note that A* is essentially positive and has the Perron-Frobenius eigenvalue λmax(A*)=-2.6456. Theorem 1(H) shows that the Hurwitz stability of A* ensures the Hurwitz diagonal stability of the polytope 𝒜 relative to any p-norm, 1≤p≤∞.
The stability margins of the polytope𝒜 are given by relation (16-H) in Theorem 2, i.e. ρHDSp(𝒜)=ρHS(𝒜)=ρHS(A*)=|λmax(A*)|=2.6456, for any p, 1≤p≤∞.
For the qualitative analysis of the continuous-time polytopic system defined by(3-H) and (24), we can apply Theorem 4(H). Remark 6(ii) shows that for any p, 1≤p≤∞, the fastest decreasing rate for the diagonal Lyapunov functions and for the contractive invariant sets is exactly λmax(A*)=-2.6456, since A* is irreducible. We apply Case 1 of the procedure presented in Remark 6(ii) and relying on the right and left Perron eigenvectors of A* (v=[10.5562]T and w=[0.78221]T), we construct the diagonal matrices Dp≻0 corresponding to the fastest decreasing rate. For p∈{1,2,∞}, these diagonal matrices are D1=diag{1.2785,1}, D2=diag{1.1307,0.7458}, and D∞=diag{1,0.5562}, and they satisfy Theorem 4(H) with r=λmax(A*)=-2.6456.
Note that all the above results remain valid if instead of𝒜 defined by (24), we consider the matrix polytope
(27)𝒜′={A∈ℝ2×2∣A-≤A≤A*},A*=[-3.631.770.77-4.03],A-=[abcd],a≤-3.63,|b|≤1.77,|c|≤0.77,d≤-4.03,
which has the same dominant vertex A* (26).


## 5. Conclusions

The paper provides analysis instruments for the stability of matrix polytopes with a dominant vertex, as well as for the dynamics of discrete- and continuous-time uncertain systems defined by such polytopes. These analysis instruments are formulated as necessary and sufficient conditions exclusively based on the characteristics of the dominant vertex. Thus, the dominant vertex represents the only test matrix used for studying the following properties of a matrix polytope and its associated dynamical system: (i) Schur (resp.,  Hurwitz) stability (including the computation of the corresponding margin); (ii) Schur (resp., Hurwitz) diagonal stability relative to ap-norm (including the computation of the corresponding margin); (iii) existence of diagonal positive-definite matrices solving the Stein-type  (resp., Lyapunov-type) inequalities relative to a p-norm; (iv) existence of diagonal-type Lyapunov functions and contractive invariant sets defined by a p-norm and decreasing with a given rate. A global result of our work is the proof that stability and diagonal stability relative to an arbitrary p-norm are equivalent for the considered class of matrix polytopes (fact which is not true for general matrix polytopes).



---

*Source: 396759-2013-06-05.xml*