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

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 Hölder 𝑝-norm, 1≤𝑝≤∞, abbreviated as SDS_𝑝 (resp., HDS_𝑝). 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 SDS_𝑝 (resp., HS and HDS_𝑝), 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.

1. Introduction

1.1. Research Context and Objective

Consider the matrix polytope 𝒜=𝐀∈ℝ^𝑛×𝑛∣𝐀=^𝐾_𝑘=1𝛾_𝑘𝐀_𝑘,𝛾_𝑘≥0,^𝐾_𝑘=1𝛾_𝑘=1, (1) where {𝐀₁,𝐀₂,…,𝐀_𝐾} 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 ℝ^𝑛×𝑛, i.e. 𝒜^𝐼=𝐀∈ℝ^𝑛×𝑛∣𝐀₀−𝐑≤𝐀≤𝐀₀,𝐀+𝐑₀,𝐑∈ℝ^𝑛×𝑛,𝐑≥0, (2) 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 on a 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 SDS_𝑝 (resp., HDS_𝑝). For 𝑝=2, the framework proposed by [28] coincides with the classic approach presented by [27].

The SDS_𝑝 (resp., HDS_𝑝), 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, 𝐱(𝑡+1)=𝐀𝐱(𝑡),𝐀∈𝒜,𝐱𝑡₀=𝐱₀,𝑡,𝑡₀∈ℤ₊,𝑡≥𝑡₀, (3-S) (ii) in the continuous-time case, 𝐱̇(𝑡)=𝐀𝐱(𝑡),𝐀∈𝒜,𝐱𝑡₀=𝐱₀,𝑡,𝑡₀∈ℝ₊,𝑡≥𝑡₀. (3-H)

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 SDS_𝑝 (resp., HDS_𝑝), 1≤𝑝≤∞, unlike the case of an arbitrary matrix polytope (e.g., [30]) where (i) SDS_𝑝 (resp., HDS_𝑝) is more conservative than SS (resp., HS) and (ii) results on SDS_𝑝1 and SDS_𝑝2 (resp., HDS_𝑝1 and HDS_𝑝2) 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

Let 𝐱=[𝑥₁⋯𝑥_𝑛]^𝑇, 𝐲=[𝑦₁⋯𝑦_𝑛]^𝑇∈ℝ^𝑛 be vectors.(i)‖𝐱‖_𝑝 is the Hölder vector 𝑝-norm defined by ‖𝐱‖_𝑝=[|𝑥₁|^𝑝+⋯+|𝑥_𝑛|^𝑝]^1/𝑝 for 1≤𝑝<∞ and by ‖𝐱‖_∞=max_{1≤𝑖≤𝑛}|𝑥_𝑖| for 𝑝=∞.(ii) “𝐱≤𝐲”, “𝐱<𝐲” mean componentwise inequalities, i.e.  𝑥_𝑖≤𝑦_𝑖, 𝑥_𝑖<𝑦_𝑖, 𝑖=1,…,𝑛.

Let 𝐌=[𝑚_𝑖𝑗], 𝐐=[𝑞_𝑖𝑗]∈ℝ^𝑛×𝑛 be square matrices.(iii)  ‖𝐌‖_𝑝 is the matrix norm induced by the vector 𝑝-norm through ‖𝐌‖_𝑝=sup_𝑥≠0(‖𝐌𝐱‖_𝑝/‖𝐱‖_𝑝)=max_{‖𝐱‖𝑝=1}‖𝐌𝐱‖_𝑝.(iv)  𝜇_𝑝(𝐌)=lim_ℎ↓0ℎ⁻¹(‖𝐼+ℎ𝐌‖_𝑝−1) is the matrix measure [36, page 41], based on the matrix norm ‖‖_𝑝.(v) If 𝐃=diag{𝑑₁,…,𝑑_𝑛}, 𝑑_𝑖>0, 𝑖=1,…,𝑛, then the following regions of the complex plane 𝐺^𝑐_𝑗(𝐃⁻¹𝐌𝐃)={𝑧∈ℂ∣|𝑧−𝑚_𝑗𝑗|≤∑^𝑛_{𝑖=1,𝑖≠𝑗}((𝑑_𝑗/𝑑_𝑖)|𝑚_𝑖𝑗|)}, 𝑗=1,…,𝑛, are called the generalized Gershgorin's disks of 𝐌 defined with 𝐃 for columns.(vi) If 𝐃=diag{𝑑₁,…,𝑑_𝑛}, 𝑑_𝑖>0, 𝑖=1,…,𝑛, then the following regions of the complex plane 𝐺^𝑟_𝑖(𝐃⁻¹𝐌𝐃)={𝑧∈ℂ∣|𝑧−𝑚_𝑖𝑖|≤∑^𝑛_{𝑗=1,𝑗≠𝑖}((𝑑_𝑗/𝑑_𝑖)|𝑚_𝑖𝑗|)}, 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 𝜆_𝑖(𝐌)∈𝜎(𝐌), 𝑖=1,…,𝑛, are the eigenvalues of 𝐌.(viii) If 𝜎(𝐌)⊂ℂ_𝑆={𝑧∈ℂ∣|𝑧|<1}, then matrix 𝐌 is said to be Schur stable (abbreviated as SS).(ix) If 𝜎(𝐌)⊂ℂ_𝐻={𝑧∈ℂ∣Re𝑧<0}, then matrix 𝐌 is said to be Hurwitz stable (abbreviated as HS).(x) If 𝐌 is nonnegative (all entries are nonnegative), its spectral radius is a positive eigenvalue, denoted by 𝜆_max(𝐌), such that |𝜆_𝑖(𝐌)|≤𝜆_max(𝐌), 𝑖=1,…,𝑛.(xi) If 𝐌 is essentially nonnegative (all off-diagonal entries are nonnegative), then it has a real eigenvalue, denoted by 𝜆_max(𝐌), such that Re{𝜆_𝑖(𝐌)}≤𝜆_max(𝐌), 𝑖=1,…,𝑛—for example, Lemma 1 in [28].(xii)If 𝐌 is symmetrical, then all its eigenvalues are real and there exists an eigenvalue denoted by 𝜆_max(𝐌), such that 𝜆_𝑖(𝐌)≤𝜆_max(𝐌), 𝑖=1,…,𝑛.(xiii) “𝐌≻0", “𝐌≺0" mean that 𝐌 is a positive-definite, negative-definite matrix.(xiv) If the oriented graph of 𝐌 is strongly connected, then 𝐌 is called irreducible; otherwise 𝐌 is called reducible.(xv) For 𝑝∈{1,2,∞}, the matrix norms ‖𝐌‖_𝑝 and matrix measures 𝜇_𝑝(𝐌) have the following expressions: ‖𝐌‖_𝑝=max_{1≤𝑗≤𝑛}^𝑛_𝑖=1𝑚_𝑖𝑗,𝑝=1,𝜆_max𝐌^𝑇𝐌,𝑝=2,max_{1≤𝑖≤𝑛}^𝑛_𝑗=1𝑚_𝑖𝑗,𝑝=∞,𝜇_𝑝(𝐌)=max_{1≤𝑗≤𝑛}𝑚_𝑗𝑗+^𝑛_{𝑖=1,𝑖≠𝑗}𝑚_𝑖𝑗1,𝑝=1,2𝜆_max𝐌+𝐌^𝑇,𝑝=2,max_{1≤𝑖≤𝑛}𝑚_𝑖𝑖+^𝑛_{𝑗=1,𝑗≠𝑖}𝑚_𝑖𝑗,𝑝=∞. (4-S)(4-H)(xvi)  |𝐌| denotes the matrix built with the absolute values of the entries of 𝐌.(xvii)  𝐌^𝑆∈ℝ^𝑛×𝑛 (𝑆 superscript from Schur) denotes the nonnegative matrix defined by 𝐌^𝑆=|𝐌|.(xviii)  𝐌^𝐻∈ℝ^𝑛×𝑛 (𝐻 superscript from Hurwitz) denotes the essentially nonnegative matrix defined by 𝐌^𝐻=𝐌^𝑑+|𝐌^𝑜|, where 𝐌^𝑑=diag{𝑚₁₁,...,𝑚_𝑛𝑛} and 𝐌^𝑜=𝐌−𝐌^𝑑.(xix)  “𝐌≤𝐐”, “𝐌<𝐐” mean componentwise inequalities, i.e. 𝑚_𝑖𝑗≤𝑞_𝑖𝑗, 𝑚_𝑖𝑗<𝑞_𝑖𝑗, 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 Hölder 𝑝-norms 1≤𝑝≤∞.

2.1. Preliminaries

Definition 1 (S). Let us consider: 1≤𝑝≤∞; an arbitrary matrix 𝐀∈ℝ^𝑛×𝑛; a matrix polytope 𝒜 of form (1); a nonsingular matrix 𝐐∈ℝ^𝑛×𝑛.
(a) The inequality 𝐐⁻¹𝐀𝐐_𝑝<1 (5-S) 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) Matrix 𝐐 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: ∀𝐀∈𝒜∶𝐐⁻¹𝐀𝐐_𝑝<1. (6-S)

Definition 1 (H). Let us consider: 1≤𝑝≤∞; an arbitrary matrix 𝐀∈ℝ^𝑛×𝑛; a matrix polytope 𝒜 of form (1); a nonsingular matrix 𝐐∈ℝ^𝑛×𝑛.
(a) The inequality 𝜇_𝑝𝐐⁻¹𝐀𝐐<0 (5-H) 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) Matrix 𝐐 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: ∀𝐀∈𝒜∶𝜇_𝑝𝐐⁻¹𝐀𝐐<0. (6-H)

Remark 1. (i) The terminology introduced by Definition 1(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 𝐀^𝑇𝐏𝐀−𝐏≺0,𝐏=𝐐^−1𝑇𝐐⁻¹, (7-S) respectively the standard Lyapunov inequality𝐀^𝑇𝐏+𝐏𝐀≺0,𝐏=𝐐^−1𝑇𝐐⁻¹. (7-H)
Indeed, for (5-S) with 𝑝=2 we may write that ‖𝐐⁻¹𝐀𝐐‖₂<1 is equivalent to ∀𝑖=1,…,𝑛∶𝜆_𝑖𝐐^𝑇𝐀^𝑇𝐐^−1𝑇𝐐⁻¹𝐀𝐐<1⟺𝜆_𝑖𝐐^𝑇𝐀^𝑇𝐐^−1𝑇𝐐⁻¹𝐀𝐐−𝐈<0⟺𝜆_𝑖𝐀^𝑇𝐐^−1𝑇𝐐⁻¹𝐀−𝐐^−1𝑇𝐐⁻¹<0, (8-S) 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 ∀𝑖=1,…,𝑛∶𝜆_𝑖𝐐^𝑇𝐀^𝑇𝐐^−1𝑇+𝐐⁻¹𝐀𝐐<0⟺𝜆_𝑖𝐀^𝑇𝐐^−1𝑇𝐐⁻¹+𝐐^−1𝑇𝐐⁻¹𝐀<0, (8-H) which means the fulfillment of (7-H).
(ii) The existence of 𝐏≻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). Let 𝒜 be a matrix polytope of form (1).
(a) 𝒜 is called Schur stable (abbreviated as SS) if ∀𝐀∈𝒜∶𝐀isSS. (9-S)
(b) Let 1≤𝑝≤∞. 𝒜 is called Schur diagonally stable relative to the 𝑝-norm (abbreviated as SDS_𝑝) 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. ∀𝐀∈𝒜∶𝐃⁻¹𝐀𝐃_𝑝<1. (10-S)

Definition 2 (H). Let 𝒜 be a matrix polytope of form (1).
(a) 𝒜 is called Hurwitz stable (abbreviated as HS) if ∀𝐀∈𝒜∶𝐀isHS. (9-H)
(b) Let 1≤𝑝≤∞. 𝒜 is called Hurwitz diagonally stable relative to the 𝑝-norm (abbreviated as HDS_𝑝) 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. ∀𝐀∈𝒜∶𝜇_𝑝𝐃⁻¹𝐀𝐃<0. (10-H)

Remark 2. (i) If 𝒜 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) Let 1≤𝑝≤∞. If 𝒜 is a proper polytope (i.e. ∃𝑘₁≠𝑘₂, 𝑘₁,𝑘₂∈{1,…,𝐾}: 𝐀_𝑘1≠𝐀_𝑘2 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) Let 1≤𝑝≤∞. The SDS_𝑝 (resp., HDS_𝑝) is a property of 𝒜 stronger than SS (resp., HS). Indeed, each matrix 𝐀∈𝒜 is SS (resp., HS) once it is SDS_𝑝 (resp., HDS_𝑝) in accordance with Remark 2 in [28].
(iv) Let 𝑝=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). Let 𝒜 be a matrix polytope of form (1). If there exists a subscript 𝑘^∗∈{1,…,𝐾}, such that the vertex 𝐀_𝑘^∗ fulfills one of the following two sets of componentwise inequalities: 𝐀^𝑆_𝑘≤𝐀_𝑘^∗,𝑘≠𝑘^∗,𝑘=1,…,𝐾,𝐀^𝑆_𝑘≤−𝐀_𝑘^∗,𝑘≠𝑘^∗,𝑘=1,…,𝐾, (11-S-1)(11-S-2) then 𝒜 is called a matrix polytope with an S-dominant vertex, and 𝐀_𝑘^∗ is called the S-dominant vertex of 𝒜. (The notation 𝐌^𝑆 is introduced in Section 1.3.)

Definition 3 (H). If there exists a subscript 𝑘^∗∈{1,…,𝐾}, such that the vertex 𝐀_𝑘^∗ fulfills the componentwise inequalities: 𝐀^𝐻_𝑘≤𝐀_𝑘^∗,𝑘≠𝑘^∗,𝑘=1,…,𝐾, (11-H) then 𝒜 is called a matrix polytope with an H-dominant vertex, and 𝐀_𝑘^∗ is called the H-dominant vertex of 𝒜. (The notation 𝐌^𝐻 is introduced in Section 1.3.)

Remark 3. (i) If 𝒜 is a matrix polytope with an S-dominant (resp., H-dominant) vertex 𝐀_𝑘^∗, then Definition 3(S) (resp., Definition 3(H)) shows that matrix 𝐀_𝑘^∗ 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 𝐀_𝑘^∗ nonnegative remain valid for −𝐀_𝑘^∗ nonnegative.
(iii) A matrix polytope 𝒜 may have two S-dominant vertices denoted as 𝐀_𝑘^∗,𝐀_𝑘^∗∗ in the particular case when 𝐀_𝑘^∗=−𝐀_𝑘^∗∗, 𝐀_𝑘^∗≥0 satisfies inequalities (11-S-1), 𝐀_𝑘^∗∗≤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 𝐀_𝑘^∗ and by 𝐀_𝑘^∗∗ 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 𝐀_𝑘^∗.
The following statements are equivalent.(i)𝐀_𝑘^∗ is SS.(ii)𝒜 is SS.(iii)There exists a 𝑝, 1≤𝑝≤∞, such that 𝒜 is SDS_𝑝.(iv)𝒜 is SDS_𝑝 for all 𝑝, 1≤𝑝≤∞.(v)There exists a diagonal matrix 𝐃≻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⋃^𝑛_𝑗=1𝐺^𝑐_𝑗(𝐃⁻¹𝐀𝐃)⊂ℂ_𝑆.(vi)There exists a diagonal matrix 𝐃≻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⋃^𝑛_𝑖=1𝐺^𝑟_𝑖(𝐃⁻¹𝐀𝐃)⊂ℂ_𝑆.

Proof. See  the Appendix.

Theorem 1 (H). Let us consider: 1≤𝑝≤∞; a matrix polytope 𝒜 with an H-dominant vertex 𝐀_𝑘^∗.
The following statements are equivalent.(i)𝐀_𝑘^∗ is HS.(ii)𝒜 is HS.(iii)There exists a 𝑝, 1≤𝑝≤∞, such that 𝒜 is HDS_𝑝.(iv)𝒜 is HDS_𝑝 for all 𝑝, 1≤𝑝≤∞.(v)There exists a diagonal matrix 𝐃≻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⋃^𝑛_𝑗=1𝐺^𝑐_𝑗(𝐃⁻¹𝐀𝐃)⊂ℂ_H.(vi)There exists a diagonal matrix 𝐃≻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⋃^𝑛_𝑖=1𝐺^𝑟_𝑖(𝐃⁻¹𝐀𝐃)⊂ℂ_𝐻.

Proof. See the Appendix.

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

The S-dominant (resp., H-dominant) vertex 𝐀_𝑘^∗ of a matrix polytope 𝒜 can be used not only for testing the properties SDS_𝑝 (resp., HDS_𝑝) 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 𝐀_𝑘^∗; a diagonal positive-definite matrix 𝐃≻0.
Matrix 𝐃 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 𝐃⁻¹𝐀_𝑘^∗𝐃_𝑝<1 (12-S) (i.e. 𝐃 satisfies the Stein-type inequality relative to the 𝑝-norm associated with the S-dominant vertex 𝐀_𝑘^∗).

Proof. See the Appendix.

Theorem 2 (H). Let us consider: 1≤𝑝≤∞; a matrix polytope 𝒜 with an H-dominant vertex 𝐀_𝑘^∗; a diagonal positive-definite matrix   𝐃≻0.
Matrix 𝐃 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 𝜇_𝑝𝐃⁻¹𝐀_𝑘^∗𝐃<0 (12-H) (i.e. 𝐃 satisfies the Lyapunov-type inequality relative to the 𝑝-norm associated with the H-dominant vertex 𝐀_𝑘^∗).

Proof. See the Appendix.

Remark 4. Let 1≤𝑝≤∞. Whenever 𝐀_𝑘^∗ 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) vertex 𝐀_𝑘^∗ of a matrix polytope 𝒜 also allows one to develop a robustness analysis for SS and SDS_𝑝 of 𝒜 defined by (1) and (11-S-1) or (11-S-2) (resp., HS and HDS_𝑝 of 𝒜 defined by (1) and (11-H)).

Definition 4 (S). Let 𝒜 be a matrix polytope with an S-dominant vertex 𝐀_𝑘^∗.
(a) If 𝐀_𝑘^∗ is SS, then 𝜌_SS𝐀_𝑘^∗=1−max_{𝑖=1,…,𝑛}𝜆_𝑖𝐀_𝑘^∗=1−𝜆_max𝐀_𝑘^∗ (13-S) is called the SS margin of 𝐀_𝑘^∗.
(b) If 𝒜 is SS, then 𝜌_SS(𝒜)=1−max_𝐴∈𝒜max_{𝑖=1,…,𝑛}𝜆_𝑖(𝐴) (14-S) is called the SS margin of 𝒜.
(c) Let 1≤𝑝≤∞. If 𝒜 is SDS_𝑝, then 𝜌_SDS𝑝(𝒜)=1−inf_{𝐃≻0diagonal}max_𝐀∈𝒜𝐃⁻¹𝐀𝐃_𝑝 (15-S) is called the SDS_𝑝 margin of 𝒜.

Definition 4 (H). Let 𝒜 be a matrix polytope with an H-dominant vertex 𝐀_𝑘^∗.
(a) If 𝐀_𝑘^∗ is HS, then 𝜌_HS𝐀_𝑘^∗=max_{𝑖=1,…,𝑛}Re𝜆_𝑖𝐀_𝑘^∗=𝜆_max𝐀_𝑘^∗ (13-H) is called the HS margin of 𝐀_𝑘^∗.
(b) If 𝒜 is HS, then 𝜌_HS(𝒜)=max_𝐀∈𝒜max_{𝑖=1,…,𝑛}Re𝜆_𝑖(𝐀) (14-H) is called the HS margin of 𝒜.
(c) Let 1≤𝑝≤∞. If 𝒜 is HDS_𝑝, then 𝜌_HDS𝑝(𝒜)=inf_{𝐃≻0diagonal}max_𝐀∈𝒜𝜇_𝑝𝐃⁻¹𝐀𝐃 (15-H) is called the HDS_𝑝 margin of 𝒜.

Theorem 3 (S). Let 𝒜 be a matrix polytope with an S-dominant vertex 𝐀_𝑘^∗. For any 𝑝, 1≤𝑝≤∞, the following equalities hold: 𝜌_SDS𝑝(𝒜)=𝜌_SS(𝒜)=𝜌_SS𝐀_𝑘^∗. (16-S)

Proof. See the Appendix.

Theorem 3 (H). Let 𝒜 be a matrix polytope with an H-dominant vertex 𝐀_𝑘^∗. For any 𝑝, 1≤𝑝≤∞, the following equalities hold: 𝜌_HDS𝑝(𝒜)=𝜌_HS(𝒜)=𝜌_HS𝐀_𝑘^∗. (16-H)

Proof. See the Appendix.

Remark 5. (i) For each stability property of the polytope 𝒜 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 SDS_p (resp., HS and HDS_𝑝) is defined by the S-dominant (resp., H-dominant) vertex.
(ii) For 𝑝=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⋃^𝑛_𝑗=1𝐺^𝑐_𝑗(𝐃⁻¹𝐀𝐃) is located in the region of the complex plane defined by |𝑧|≤1−𝜌_SS(𝐀_𝑘^∗) (resp., Re𝑧≤−𝜌_SS(𝐀_𝑘^∗)). The same location also corresponds to the union for all 𝐀∈𝒜 of the generalized Gershgorin’s disks written for rows ⋃_𝐀∈𝒜⋃^𝑛_𝑖=1𝐺^𝑟_𝑖(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−𝜌_SS(𝐀_𝑘^∗) (resp., Re𝑧≤−𝜌_SS(𝐀_𝑘^∗)) 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 polytope 𝒜 (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 SDS_𝑝 (resp., HDS_𝑝) then 𝜌_SDS𝑝(𝒜)≤𝜌_SS(𝒜) (resp., 𝜌_HDS𝑝(𝒜)≤𝜌_HS(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 𝜌_SDS𝑝1(𝒜)≠𝜌_SDS𝑝2(𝒜) (resp., 𝜌_HDS𝑝1(𝒜)≠𝜌_HDS𝑝2(𝒜)), as already suggested by our recent paper [30].

2.5. Particular Case of Interval Matrices with a Dominant Vertex

Theorems 1(S), 2(S), and 3(S) generalize the results reported in [27, Lemma 3.4.18], [29] for SS and SDS_𝑝 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, Theorems 1(H), 2(H), and 3(H) generalize results reported in [29] for HS and HDS_𝑝 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 𝐀_𝑘^∗; a positive-definite diagonal matrix    𝐃≻0; a constant   𝑟 satisfying 0<𝑟<1.
The following statements are equivalent:(i) 𝐃⁻¹𝐀_𝑘^∗𝐃_𝑝≤𝑟. (17-S)(ii) For the polytopic system (3-S), the functions 𝒱^𝛼_𝑝∶ℝ^𝑛⟶ℝ₊,𝒱^𝛼_𝑝(𝐱)=1𝛼𝐃⁻¹𝐱_𝑝,𝛼>0 (18-S) are strong diagonal Lyapunov functions, with the decreasing rate 𝑟, i.e. ∀𝐱(𝑡)𝑖𝑠𝑎𝑡𝑟𝑎𝑗𝑒𝑐𝑡𝑜𝑟𝑦𝑜𝑓𝑠𝑦𝑠𝑡𝑒𝑚(3-S),∀𝑡∈ℤ₊,𝑡≥𝑡₀∶𝒱^𝛼_𝑝(𝐱(𝑡+1))≤𝑟𝒱^𝛼_𝑝(𝐱(𝑡)). (19-S)(iii) The contractive sets 𝒳^𝛼_𝑝𝑡;𝑡₀=𝐱∈ℝ^𝑛∣𝐃⁻¹𝐱_𝑝≤𝛼𝑟^{(𝑡−𝑡0)},𝑡,𝑡₀∈ℤ₊,𝑡≥𝑡₀,𝛼>0 (20-S) are invariant with respect to the state-space trajectories (solutions) of the polytopic system (3-S), i.e. ∀𝛼>0,∀𝑡,𝑡₀∈ℤ₊,𝑡≥𝑡₀,∀𝐱₀=𝐱𝑡₀∈ℝ^𝑛∶𝐃⁻¹𝐱_0𝑝≤𝛼⟹𝐃⁻¹𝐱𝑡;𝑡₀,𝐱_0𝑝≤𝛼𝑟^{(𝑡−𝑡0)}. (21-S)

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 𝐀_𝑘^∗; a positive-definite diagonal matrix   𝐃≻0; a constant   𝑟 satisfying 𝑟<0.
The following statements are equivalent:(i) 𝜇_𝑝𝐃⁻¹𝐀_𝑘^∗𝐃≤𝑟. (17-H)(ii) For the polytopic system (3-H), the functions 𝒱^𝛼_𝑝∶ℝ^𝑛⟶ℝ₊,𝒱^𝛼_𝑝(𝐱)=1𝛼𝐃⁻¹𝐱_𝑝,𝛼>0 (18-H) are strong diagonal Lyapunov functions, with the decreasing rate 𝑟, i.e. ∀𝐱(𝑡)𝑖𝑠𝑎𝑡𝑟𝑎𝑗𝑒𝑐𝑡𝑜𝑟𝑦𝑜𝑓𝑠𝑦𝑠𝑡𝑒𝑚(3-H),∀𝑡∈ℝ₊,𝑡≥𝑡₀∶𝐷⁺𝒱^𝛼_𝑝(𝐱(𝑡))=lim_ℎ↓01ℎ𝒱^𝛼_𝑝(𝐱(𝑡+ℎ))−𝒱^𝛼_𝑝(𝐱(𝑡))≤𝑟𝒱^𝛼_𝑝(𝐱(𝑡)). (19-H)(iii) The contractive sets 𝒳^𝛼_𝑝𝑡;𝑡₀=𝐱∈ℝ^𝑛∣𝐃⁻¹𝐱_𝑝≤𝛼𝑒^{𝑟(𝑡−𝑡0)},𝑡,𝑡₀∈ℝ₊,𝑡≥𝑡₀,𝛼>0 (20-H) are invariant with respect to the state-space trajectories (solutions) of the polytopic system (3-H), i.e. ∀𝛼>0,∀𝑡,𝑡₀∈ℝ₊,𝑡≥𝑡₀,∀𝐱₀=𝐱𝑡₀∈ℝ^𝑛∶𝐃⁻¹𝐱_0𝑝≤𝛼⟹𝐃⁻¹𝐱𝑡;𝑡₀,𝐱₀≤𝛼𝑒^{𝑟(𝑡−𝑡0)}. (21-H)

Proof. See the Appendix.

Remark 6. (i) Let 1≤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 SDS_𝑝 (resp., HDS_𝑝) 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 constant 0<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 𝜆_max(𝐀_𝑘^∗), regardless of the discrete-time or continuous-time nature of the dynamics. Indeed, for 𝑟<𝜆_max(𝐀_𝑘^∗), there exists no diagonal matrix 𝐃≻0 satisfying inequality (17-S) (resp., (17-H)). For 𝑟≥𝜆_max(𝐀_𝑘^∗), we use Lemma 3 and Remark 3 in [28] that yield the following discussion on the irreducibility/reducibility of 𝐀_𝑘^∗.

Case 1 (matrix 𝐀_𝑘^∗ is irreducible). (The definition of an irreducible matrix is introduced in Section 1.3.) Denote by 𝐯=[𝑣₁⋯𝑣_𝑛]^𝑇>0 and 𝐰=[𝑤₁⋯𝑤_𝑛]^𝑇>0 its right and left Perron eigenvectors, respectively. Given 𝑝, 1≤𝑝≤∞, we construct the diagonal matrix 𝐃_𝑝=diag{𝑣₁^1/𝑞/𝑤₁^1/𝑝,…,𝑣_𝑛^1/𝑞/𝑤_𝑛^1/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 𝐃_𝑝≻0 fulfills the following equality:(i)‖𝐃_𝑝⁻¹𝐀_𝑘^∗𝐃_𝑝‖_𝑝=𝜆_max(𝐀_𝑘^∗) if 𝐀_𝑘^∗ is nonnegative, i.e. 𝐀_𝑘^∗ is defined by (11-S-1); (ii)‖𝐃_𝑝⁻¹𝐀_𝑘^∗𝐃_𝑝‖_𝑝=𝜆_max(−𝐀_𝑘^∗) if 𝐀_𝑘^∗ is nonpositive, i.e. 𝐀_𝑘^∗ is defined by (11-S-2); (iii)𝜇_𝑝(𝐃_𝑝⁻¹𝐀_𝑘^∗𝐃_𝑝)=𝜆_max(𝐀_𝑘^∗) if 𝐀_𝑘^∗ is essentially nonnegative, i.e. 𝐀_𝑘^∗ is defined by (11-H).

Case 2 (matrix 𝐀_𝑘^∗ is reducible). For any 𝑟>𝜆_max(𝐀_𝑘^∗), there exists 𝜀>0 such that 𝜆_max(𝐀_𝑘^∗)<𝜆_max(𝐀_𝑘^∗+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 𝐀_𝑘^∗+𝜀𝐉, and the resulting diagonal matrix 𝐃_𝑝≻0 fulfills ‖𝐃_𝑝⁻¹(𝐀_𝑘^∗+𝜀𝐉)𝐃_𝑝‖_𝑝=𝜆_max(𝐀_𝑘^∗+𝜀𝐉)<𝑟, 𝜇_𝑝(𝐃_𝑝⁻¹(𝐀_𝑘^∗+𝜀𝐉)𝐃_𝑝)=𝜆_max(𝐀_𝑘^∗+𝜀𝐉)<𝑟, respectively. On the other hand, we have the norm inequality ‖𝐃_𝑝⁻¹𝐀_𝑘^∗𝐃_𝑝‖_𝑝≤‖𝐃_𝑝⁻¹(𝐀_𝑘^∗+𝜀𝐉)𝐃_𝑝‖_𝑝, the measure inequality 𝜇_𝑝(𝐃_𝑝⁻¹𝐀_𝑘^∗𝐃_𝑝)≤𝜇_𝑝(𝐃_𝑝⁻¹(𝐀_𝑘^∗+𝜀𝐉)𝐃_𝑝), respectively. Subsequently, we obtain 𝜆_max(𝐀_𝑘^∗)<‖𝐃_𝑝⁻¹𝐀_𝑘^∗𝐃_𝑝‖_𝑝<𝑟, 𝜆_max(𝐀_𝑘^∗)<𝜇_𝑝(𝐃_𝑝⁻¹𝐀_𝑘^∗𝐃_𝑝)<𝑟, respectively.
Thus, for any 𝑝, 1≤𝑝≤∞, we can construct a diagonal matrix 𝐃_𝑝≻0 such that inequality (17-S) (resp., (17-H)) is fulfilled with 𝑟=𝜆_max(𝐀_𝑘^∗)—for 𝐀_𝑘^∗ irreducible and with 𝑟>𝜆_max(𝐀_𝑘^∗) but as close to 𝜆_max(𝐀_𝑘^∗) as we want—for 𝐀_𝑘^∗ 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 Section 2.4), as 1−𝜌_SS(𝐀_𝑘^∗) for the discrete-time case and −𝜌_SS(𝐀_𝑘^∗) 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 polytope 𝒜 (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 SDS_𝑝 (resp., HDS_𝑝) 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−𝜌_SDS𝑝(𝒜) (resp., −𝜌_HDS𝑝(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. Example 1 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 𝒮^[−1,1]=𝐒∈ℝ^𝑛×𝑛∣𝐒=diag𝑠₁,…,𝑠_𝑛,𝑠_𝑖=≤1,𝑖=1,…,𝑛𝐒∈ℝ^𝑛×𝑛∣|𝐒|≤𝐈_𝑛 (22) be the set of diagonal matrices whose elements are subunitary. 𝒮^[−1,1] is the convex hull generated by the set of 2^𝑛 vertices 𝒮_sgn={𝐒∈ℝ^𝑛×𝑛∣|𝐒|=𝐈_𝑛}, also called the class of signature matrices.
Given a nonnegative matrix 𝐀^∗∈ℝ^𝑛×𝑛, the set 𝒜=ℛ𝐀^∗=𝐀=𝐀^∗𝐒∣𝐒∈𝒮^[−1,1] (23) is a matrix polytope of form (1) with the vertices 𝐀_𝑖=𝐀^∗𝐒_𝑖, where 𝐒_𝑖∈𝒮_sgn, 𝑖=1,…,2^𝑛. 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).
Theorem 1(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 𝐃_𝑝≻0 satisfies the Stein-type inequality relative to the 𝑝-norm associated with 𝒜=ℛ(𝐀^∗) if and only if 𝐃_𝑝≻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.
If 𝐀^∗ is Schur stable, then the eigenvalue 𝜆_max(𝐀^∗) allows one to investigate the following properties: (i) for the polytope 𝒜=ℛ(𝐀^∗), the SDS_𝑝 margins are given by relation (16-S) in Theorem 3(S) i.e. 𝜌_SDS𝑝(ℛ(𝐀^∗))=1−𝜆_max(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 𝜆_max(𝐀^∗) if 𝐀^∗ is irreducible and arbitrarily close to 𝜆_max(𝐀^∗) if 𝐀^∗ is reducible—as per Remark 6(ii).
Finally, we notice that the nonpositive matrix 𝐀^∗∗=−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 𝜆_max(−𝐀^∗∗)=𝜆_max(𝐀^∗), and for all 𝑝, 1≤𝑝≤∞, the equality ‖𝐃⁻¹𝐀^∗∗𝐃‖_𝑝=‖𝐃⁻¹𝐀^∗𝐃‖_𝑝 holds true).
The above approach applies mutatis-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]: 𝒜=𝐀∈ℝ^2×2∣𝐀=𝐀₀|Δ|,𝐀+Δ,≤𝐑₀=,−3.81.60.6−4.2,𝐑=0.171111 (24) that is a matrix polytope with 𝐾=2⁴ vertices: 𝐀_𝑘=𝐀₀+Δ_𝑘,𝑘=1,…,16,whereΔ_𝑘=0.17.±1±1±1±1 (25)𝒜 has an H-dominant vertex 𝐀^∗=𝐀₀+𝐑=,−3.631.770.77−4.03 (26) which satisfies inequalities (11-H). Note that 𝐀^∗ is essentially positive and has the Perron-Frobenius eigenvalue 𝜆_max(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 polytope 𝒜 are given by relation (16-H) in Theorem 2, i.e. 𝜌_HDS𝑝(𝒜)=𝜌_HS(𝒜)=𝜌_HS(𝐀^∗)=|𝜆_max(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 𝜆_max(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 𝐃_𝑝≻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=𝜆_max(𝐀^∗)=−2.6456.
Note that all the above results remain valid if instead of 𝒜 defined by (24), we consider the matrix polytope 𝒜^′=𝐀∈ℝ^2×2∣𝐀⁻≤𝐀≤𝐀^∗,𝐀^∗=−3.631.770.77−4.03,𝐀⁻=,|𝑏||𝑐|𝑎𝑏𝑐𝑑𝑎≤−3.63,≤1.77,≤0.77,𝑑≤−4.03, (27) 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 a 𝑝-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).

Appendix

Proof of Theorem 1(S). (i) ⇒(ii).
It results from the following: ∀𝐀∈𝒜∶𝐀=^𝐾_𝑘=1𝛾_𝑘𝐀_𝑘⟹𝐀≤𝐀^𝑆≤^𝐾_𝑘=1𝛾_𝑘𝐀^𝑆_𝑘≤^𝐾_𝑘=1𝛾_𝑘𝐀_𝑘^∗=^𝐾_𝑘=1𝛾_𝑘𝐀_𝑘^∗=𝐀_𝑘^∗⟹max_{𝑖=1,…,𝑛}𝜆_𝑖(𝐀)≤𝜆_max𝐀_𝑘^∗<1, (A.1) since 𝐀^𝑆, 𝐀_𝑘^∗ are nonnegative and we can apply Theorem 8.1.18 and Corollary 8.1.19 in [38].
(ii) ⇒(i) It is obvious, because 𝐀_𝑘^∗∈𝒜.
(i) ⇒(iv) Let 1≤𝑝≤∞. Lemma 3 in [28] ensures the existence of a diagonal matrix 𝐃≻0 such that 𝜆_max(𝐀_𝑘^∗)=𝜆_max(𝐃⁻¹𝐀_𝑘^∗𝐃)≤‖𝐃⁻¹𝐀_𝑘^∗𝐃‖_𝑝<1. On the other hand, we have the implication ∀𝐀∈𝒜∶𝐀=^𝐾_𝑘=1𝛾_𝑘𝐀_𝑘⟹𝐃⁻¹𝐀𝐃_𝑝≤𝐃⁻¹𝐀^𝑆𝐃_𝑝≤^𝐾_𝑘=1𝛾_𝑘𝐃⁻¹𝐀^𝑆_𝑘𝐃_𝑝≤^𝐾_𝑘=1𝛾_𝑘𝐃⁻¹𝐀_𝑘^∗𝐃_𝑝=^𝐾_𝑘=1𝛾_𝑘𝐃⁻¹𝐀_𝑘^∗𝐃_𝑝=𝐃⁻¹𝐀_𝑘^∗𝐃_𝑝, (A.2) as per Lemma 4 in [28]. We conclude that for all 𝐀∈𝒜∶‖𝐃⁻¹𝐀𝐃‖_𝑝<1.
(iv) ⇒(iii) It is obvious.
(iii) ⇒(ii) It follows from Remark 2(iii).
(v) ⇔(iii) with 𝑝=1. It results from the equivalence ∀𝐀∈𝒜∶𝐺^𝑐_𝑗𝐃⁻¹𝐀𝐃⊂𝐂_𝑆,𝑗=1,…,𝑛⟺−1<𝑎_𝑗𝑗−^𝑛_{𝑖=1,𝑖≠𝑗}𝑑_𝑗𝑑_𝑖𝑎_𝑖𝑗,𝑎_𝑗𝑗+^𝑛_{𝑖=1,𝑖≠𝑗}𝑑_𝑗𝑑_𝑖𝑎_𝑖𝑗⟺𝐃<1,𝑗=1,…,𝑛₋₁𝐀𝐃_𝑝<1. (A.3)
(vi) ⇔(iii) with 𝑝=∞. It is similar to (v) ⇔(iii) with 𝑝=1.

Proof of Theorem 1(H). (i) ⇒(ii).
It results from the following: ∀𝐀∈𝒜∶𝐀=^𝐾_𝑘=1𝛾_𝑘𝐀_𝑘⟹𝐀≤𝐀^𝐻≤^𝐾_𝑘=1𝛾_𝑘𝐀^𝐻_𝑘≤^𝐾_𝑘=1𝛾_𝑘𝐀_𝑘^∗=^𝐾_𝑘=1𝛾_𝑘𝐀_𝑘^∗=𝐀_𝑘^∗. (A.4) Take 𝑠>0 so that 𝑠𝐈+𝐀^𝐻≥0. Hence, |𝑠𝐈+𝐀|=𝑠𝐈+𝐀^𝐻≤𝑠𝐈+𝐀_𝑘^∗. By applying Theorem 8.1.18 and Corollary 8.1.19 in [38], we get 𝑠+max_{𝑖=1,…,𝑛}Re{𝜆_𝑖(𝐀)}≤𝑠+𝜆_max(𝐀_𝑘^∗), which implies that max_{𝑖=1,…,𝑛}Re{𝜆_𝑖(A)}≤𝜆_max(𝐀_𝑘^∗)<0.
(ii) ⇒(i) It is obvious, because 𝐀_𝑘^∗∈𝒜.
(i) ⇒(iv) Let 1≤𝑝≤∞. Lemma 3 in [28] ensures the existence of a diagonal matrix 𝐃≻0 such that 𝜆_max(𝐀_𝑘^∗)≤𝜇_𝑝(𝐃⁻¹𝐀_𝑘^∗𝐃)<0. On the other hand, we have the implication ∀𝐀∈𝒜∶𝐀=^𝐾_𝑘=1𝛾_𝑘𝐀_𝑘⟹𝜇_𝑝𝐃⁻¹𝐀𝐃≤𝜇_𝑝𝐃⁻¹𝐀^𝐻𝐃≤^𝐾_𝑘=1𝛾_𝑘𝜇_𝑝𝐃⁻¹𝐀^𝐻_𝑘𝐃≤^𝐾_𝑘=1𝛾_𝑘𝜇_𝑝𝐃⁻¹𝐀_𝑘^∗𝐃=^𝐾_𝑘=1𝛾_𝑘𝜇_𝑝𝐃⁻¹𝐀_𝑘^∗𝐃=𝜇_𝑝𝐃⁻¹𝐀_𝑘^∗𝐃, (A.5) as per Lemma 4 in [28]. We conclude that for all 𝐀∈𝒜∶𝜇_𝑝(𝐃⁻¹𝐀𝐃)<0.
(iv) ⇒(iii) It is obvious.
(iii) ⇒(ii) It follows from Remark 2(iii).
(v) ⇔(iii) with 𝑝=1. It results from the equivalence ∀𝐀∈𝒜∶𝐺^𝑐_𝑗𝐃⁻¹𝐀𝐃⊂𝐂_𝐻,𝑗=1,…,𝑛⟺𝑎_𝑗𝑗+^𝑛_{𝑖=1,𝑖≠𝑗}𝑑_𝑗𝑑_𝑖𝑎_𝑖𝑗<0,𝑗=1,…,𝑛⟺𝜇₁𝐃⁻¹𝐀𝐃<0. (A.6)
(vi) ⇔(iii) With 𝑝=∞. It is similar to (v) ⇔(iii) with 𝑝=1.

Proof of Theorem 2(S). From the proof (i) ⇒(iv) of Theorem 1(S), we can write ‖𝐃⁻¹𝐀𝐃‖_𝑝≤‖𝐃⁻¹𝐀_𝑘^∗𝐃‖_𝑝 for all 𝐀∈𝒜. Thus, if 𝐃≻0 satisfies inequality (12-S) (i.e. the Stein-type inequality relative to the 𝑝-norm associated with the matrix 𝐀_𝑘^∗), then 𝐃≻0 satisfies inequality (10-S) (i.e. the Stein-type inequality relative to the 𝑝-norm associated with the polytope 𝒜). The converse part is obvious, since 𝐀_𝑘^∗∈𝒜.

Proof of Theorem 2(H). From the proof (i) ⇒(iv) of Theorem 1(H), we can write 𝜇_𝑝(𝐃⁻¹𝐀𝐃)≤𝜇_𝑝(𝐃⁻¹𝐀_𝑘^∗𝐃) for all 𝐀∈𝒜. Thus, if 𝐃≻0 satisfies inequality (12-H) (i.e. the Lyapunov-type inequality relative to the 𝑝-norm associated with the matrix 𝐀_𝑘^∗), then 𝐃≻0 satisfies inequality (10-H) (i.e. the Lyapunov-type inequality relative to the 𝑝-norm associated with the polytope 𝒜). The converse part is obvious, since 𝐀_𝑘^∗∈𝒜.

Proof of Theorem 3(S). From the proof (i) ⇒(ii) of Theorem 1(S), we have 𝜌_SS(𝒜)≥𝜌_SS(𝐀_𝑘^∗), and from 𝐀_𝑘^∗∈𝒜, we get 𝜌_SS(𝒜)≤𝜌_SS(𝐀_𝑘^∗), such that we can conclude that 𝜌_SS(𝒜)=𝜌_SS(𝐀_𝑘^∗). Let 1≤𝑝≤∞ and 𝜀>0. Lemma 3 in [28] ensures the existence of a diagonal matrix 𝐃≻0 such that 𝜆_max(𝐀_𝑘^∗)≤‖𝐃⁻¹𝐀_𝑘^∗𝐃‖_𝑝< 𝜆_max(𝐀_𝑘^∗)+𝜀. At the same time, from the proof (i) ⇒(iv) of Theorem 1(S) we have ‖𝐃⁻¹𝐀𝐃‖_𝑝≤‖𝐃⁻¹𝐀_𝑘^∗𝐃‖_𝑝 for all 𝐀∈𝒜. As 𝐀_𝑘^∗∈𝒜, we may write inf_{𝐃≻0diagonal}max_𝐀∈𝒜‖𝐃⁻¹𝐀𝐃‖_𝑝=inf_{𝐃≻0diagonal}‖𝐃⁻¹𝐀_𝑘^∗𝐃‖_𝑝=𝜆_max(𝐀_𝑘^∗) and, subsequently, 𝜌_SDS𝑝(𝒜)=𝜌_SS(𝐀_𝑘^∗).

Proof of Theorem 3(H). From the proof (i) ⇒(ii) of Theorem 1(H), we have 𝜌_HS(𝒜)≥𝜌_HS(𝐀_𝑘^∗), and from 𝐀_𝑘^∗∈𝒜, we get 𝜌_HS(𝒜)≤𝜌_HS(𝐀_𝑘^∗), such that we can conclude that 𝜌_HS(𝒜)=𝜌_HS(𝐀_𝑘^∗). Let 1≤𝑝≤∞ and 𝜀>0. Lemma 3 in [28] ensures the existence of a diagonal matrix 𝐃≻0 such that 𝜆_max(𝐀_𝑘^∗)≤ 𝜇_𝑝(𝐃⁻¹𝐀_𝑘^∗𝐃)<𝜆_max(𝐀_𝑘^∗)+𝜀. At the same time, from the proof (i)⇒(iv) of Theorem 1(H), we have 𝜇_𝑝(𝐃⁻¹𝐀𝐃)≤𝜇_𝑝(𝐃⁻¹𝐀_𝑘^∗𝐃) for all 𝐀∈𝒜. As 𝐀_𝑘^∗∈𝒜, we may write |inf_{𝐃≻0diagonal}max_𝐀∈𝒜𝜇_𝑝(𝐃⁻¹𝐀𝐃)|=|inf_{𝐃≻0diagonal}𝜇_𝑝(𝐃⁻¹𝐀_𝑘^∗𝐃)|=|𝜆_max(𝐀_𝑘^∗)|  and, subsequently, 𝜌_HDS𝑝(𝒜)=𝜌_HS(A_k^∗).

Proof of Theorem 4(S). Inequality (17-S) is equivalent to the statement ‖𝐃⁻¹𝐀𝐃‖_𝑝≤𝑟 for all 𝐀∈𝒜. This results from the inequality ‖𝐃⁻¹𝐀𝐃‖_𝑝≤‖𝐃⁻¹𝐀_𝑘^∗𝐃‖_𝑝, for all 𝐀∈𝒜 that was obtained in the proof for (i) ⇒(iv) of Theorem 1(S). Then we apply Theorem 2 in [28] to all matrices 𝐀 in 𝒜, and we get the equivalence (17-S)  ⇔(19-S) ⇔(21-S).

Proof of Theorem 4(H). Inequality (17-H) is equivalent to the statement 𝜇_𝑝(𝐃⁻¹𝐀𝐃)≤𝑟 for all 𝐀∈𝒜. This results from the inequality 𝜇_𝑝(𝐃⁻¹𝐀𝐃)≤𝜇_𝑝(𝐃⁻¹𝐀_𝑘^∗𝐃), for all 𝐀∈𝒜 that was obtained in the proof for (i) ⇒(iv) of Theorem 1(H). Then, we apply Theorem 2 in [28] to all matrices 𝐀 in 𝒜, and we get the equivalence (17-H) ⇔(19-H)  ⇔(21-H).

Acknowledgment

The authors acknowledge the support of UEFISCDI Romania, Grant no. PN-II-ID-PCE-2011-3-1038.