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

and Applied Analysis 3 (xi) IfM is essentially nonnegative (all off-diagonal entries are nonnegative), then it has a real eigenvalue, denoted byλmax(M), such that Re{λi(M)} ≤ λmax(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 λmax(M), such that λi(M) ≤ λmax(M), i = 1, . . . , n. (xiii) “M ≻ 0”, “M ≺ 0” mean thatM 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 μp(M) have the following expressions: ‖M‖p = {{{{{{{{{ {{{{{{{{{ { max 1≤j≤n { n

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 Hölder-norm, 1 ≤  ≤ ∞, and generalized the aforementioned diagonal stability concept to "Schur (resp., Hurwitz) diagonal stability relative to a Hölder -norm"-abbreviated as SDS  (resp., HDS  ).For  = 2, the framework proposed by [28] coincides with the classic approach presented by [27].
The SDS  (resp., HDS  ), 1 ≤  ≤ ∞, 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.SDS 2 and HDS 2 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][28][29][31][32][33][34][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 ( + 1) = Ax () , A ∈ A, x ( 0 ) = x 0 , ,  0 ∈ Z + ,  ≥  0 , (3-S) (ii) in the continuous-time case, ẋ () = Ax () , A ∈ A, x ( 0 ) = x 0 , ,  0 ∈ R + ,  ≥  0 . (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 1.2.Paper Structure.For a matrix polytope (1) with a dominant vertex, we prove that SS (resp., HS) is equivalent to SDS  (resp., HDS  ), 1 ≤  ≤ ∞, unlike the case of an arbitrary matrix polytope (e.g., [30]) where (i) SDS  (resp., HDS  ) is more conservative than SS (resp., HS) and (ii) results on SDS  1 and SDS  2 (resp., HDS  1 and HDS  2 ) may be different for  1 ̸ =  2 , 1 ≤  1 ,  2 ≤ ∞.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 -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.
(xii) If M is symmetrical, then all its eigenvalues are real and there exists an eigenvalue denoted by  max (M), such that   (M) ≤  max (M),  = 1, . . ., .
(xiv) If the oriented graph of M is strongly connected, then M is called irreducible; otherwise M is called reducible.
Throughout the text we shall write "X (resp., Y)" wherever "X" and "Y" are referred to in parallel.

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 -norms 1 ≤  ≤ ∞.
(a) The inequality is called the Stein-type inequality relative to the -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 -norm associated with the polytope A if the following condition is fulfilled: is called the Lyapunov-type inequality relative to the -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 -norm associated with the polytope A if the following condition is fulfilled: 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  = 2 is equivalent to the standard Stein inequality respectively the standard Lyapunov inequality Indeed, for (5-S) with  = 2 we may write that which means the fulfillment of (7-S).
Similarly, for (5-H) with  = 2 we may write that which means the fulfillment of (7-H).
Definition 2 (S).Let A be a matrix polytope of form ( 1).
(a) A is called Schur stable (abbreviated as SS) if A is called Schur diagonally stable relative to the -norm (abbreviated as SDS  ) if there exists a diagonal positive-definite matrix D ≻ 0 that satisfies the Stein-type inequality relative to the -norm associated with the polytope A, i.e.
Definition 2 (H).Let A be a matrix polytope of form (1). (a) A is called Hurwitz stable (abbreviated as HS) if A is called Hurwitz diagonally stable relative to the -norm (abbreviated as HDS  ) if there exists a diagonal positive-definite matrix D ≻ 0 that satisfies the Lyapunov-type inequality relative to the -norm associated with the polytope A, i.e.

∀A ∈ A : 𝜇
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.

Definition 3 (S).
Let A be a matrix polytope of form (1).If there exists a subscript  * ∈ {1, . . ., }, such that the vertex A  * fulfills one of the following two sets of componentwise inequalities: S-1) (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  * nonnegative remain valid for −A  * nonnegative.
(iii) A matrix polytope A may have two S-dominant vertices denoted as A  * , A  * * in the particular case when A  * = − A  * * , A  * ≥ 0 satisfies inequalities (11-S-1), A  * * ≤ 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  * and by A  * * are identical, as resulting from the further development of our paper.

Theorem 1 (S). Let us consider: 1 ≤ 𝑝 ≤ ∞; a matrix polytope A with an S-dominant vertex
The following statements are equivalent.
(i) A  * is SS.
(ii) A is SS.
(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.
(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.
Proof.See the Appendix.
Theorem 1 (H).Let us consider: 1 ≤  ≤ ∞; a matrix polytope A with an H-dominant vertex A  * .The following statements are equivalent.
(i) A  * is HS.
(ii) A is HS.
(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.
(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.
Proof.See the Appendix.

Diagonal Solutions to Stein-Type and Lyapunov-Type
Inequalities.The S-dominant (resp., H-dominant) vertex A  * of a matrix polytope A can be used not only for testing the properties SDS  (resp., HDS  ) 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)).Proof.See the Appendix.

Stability Margins.
The S-dominant (resp., H-dominant) vertex A  * of a matrix polytope A also allows one to develop a robustness analysis for SS and SDS  of A defined by ( 1) and (11-S-1) or (11-S-2) (resp., HS and HDS  of A defined by ( 1) and (11-H)).

Definition 4 (S). Let A be a matrix polytope with an Sdominant vertex
()     (14-S) is called the SDS  margin of A. is called the HDS  margin of A.

Theorem 3 (S).
Let A be a matrix polytope with an Sdominant vertex A  * .For any , 1 ≤  ≤ ∞, the following equalities hold: Proof.See the Appendix.

Theorem 3 (H).
Let A be a matrix polytope with an Hdominant vertex A  * .For any , 1 ≤  ≤ ∞, the following equalities hold:

-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  (resp., HS and HDS  ) is defined by the S-dominant (resp., H-dominant) vertex.
(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.

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  of interval matrices of form (2) with A 0 or −A 0 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  of interval matrices of form (2) with A 0 essentially nonnegative, since such interval matrices represent a particular case of matrix polytopes with an H-dominant vertex defined by inequalities (11-H).

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.
The following statements are equivalent: (19-S) (iii) The contractive sets are invariant with respect to the state-space trajectories (solutions) of the polytopic system (3-S), i.e.
Proof.See the Appendix.
The following statements are equivalent: (ii) For the polytopic system (3-H), the functions The contractive sets (ii) The constant 0 <  < 1 (resp.,  < 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 , 1 ≤  ≤ ∞, the value of the fastest decreasing rate is given by the  max (A  * ), regardless of the discrete-time or continuous-time nature of the dynamics.Indeed, for  <  max (A  * ), there exists no diagonal matrix D ≻ 0 satisfying inequality (17-S) (resp., (17-H)).For  ≥  max (A  * ), we use Lemma 3 and Remark 3 in [28] that yield the following discussion on the irreducibility/reducibility of A  * .

Case 1 (matrix
0 its right and left Perron eigenvectors, respectively.Given , 1 ≤  ≤ ∞, we construct the diagonal matrix Case 2 (matrix A  * is reducible).For any  >  max (A  * ), there exists  > 0 such that  max (A  * ) <  max (A  * + J) < , where J is the  by  matrix with all its entries 1.We apply the procedure presented by Case 1 to the irreducible matrix A  * + J, and the resulting diagonal matrix Thus, for any , 1 ≤  ≤ ∞, we can construct a diagonal matrix D  ≻ 0 such that inequality (17-S) (resp., (17-H)) is fulfilled with  =  max (A  * )-for A  * irreducible and with  >  max (A  * ) but as close to  max (A  * ) as we want-for A  * 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 −  SS (A  * ) for the discrete-time case and − SS (A  * ) 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 -norm that defines the Lyapunov function and the invariant sets (if exist).If, for a given , 1 ≤  ≤ ∞, A is SDS  (resp., HDS  ) then the fastest decreasing rate corresponding to the -norm of the polytopic system can be expressed in terms of the stability margins as 1 −  SDS  (A) (resp., − HDS  (A)).The fastest decreasing rate corresponding to the -norm can be fairly estimated by a computational procedure based on a bisection method presented in [30].

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 Hdominant 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 be the set of diagonal matrices whose elements are subunitary. 1]is the convex hull generated by the set of 2  vertices S sgn = {S ∈ R × | |S| = I  }, also called the class of signature matrices.Given a nonnegative matrix A * ∈ R × , the set is a matrix polytope of form (1) with the vertices A  = A * S  , where S  ∈ S sgn ,  = 1, . . ., 2  .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 -norm, 1 ≤  ≤ ∞.According to Theorem 2(S), a diagonal positivedefinite matrix D  ≻ 0 satisfies the Stein-type inequality relative to the -norm associated with A = R(A * ) if and only if D  ≻ 0 satisfies the Stein-type inequality relative to the -norm associated with A * .This property of A = R(A * ) is guaranteed for any -norm by Theorem 2(S), whereas Proposition 2.5.8 in [27] can guarantee only the particular case corresponding to  = 2.
If A * is Schur stable, then the eigenvalue  max (A * ) allows one to investigate the following properties: (i) for the polytope A = R(A * ), the SDS  margins are given by relation (16-S) in Theorem 3(S) i.e.  SDS  (R(A * )) = 1 −  max (A * ), for any , 1 ≤  ≤ ∞; (ii) for the discrete-time polytopic system of form (3-S) defined by A = R(A * ), regardless of the 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  max (A * ) if A * is irreducible and arbitrarily close to  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  max (−A * * ) =  max (A * ), and for all , 1 ≤  ≤ ∞, the equality ‖D −1 A * * D‖  = ‖D −1 A * D‖  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 ≤  ≤ ∞ the result that can be obtained when  = 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]: that is a matrix polytope with  = 2 4 vertices: A has an H-dominant vertex 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 , 1 ≤  ≤ ∞, the fastest decreasing rate for the diagonal Lyapunov functions and for the contractive invariant sets is exactly  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 * (k = [1 0.5562]  and w = [0.78221]  ), we construct the diagonal matrices D  ≻ 0 corresponding to the fastest decreasing rate.For  ∈ {1, 2, ∞}, these diagonal matrices are D 1 = diag{1.2785,1}, D 2 = diag{1.1307,0.7458}, and D ∞ = diag{1, 0.5562}, and they satisfy Theorem 4(H) with  =  max (A * ) = −2.6456.
Note that all the above results remain valid if instead of A defined by (24), we consider the matrix polytope which has the same dominant vertex A * (26).

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 -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 -norm; (iv) existence of diagonal-type Lyapunov functions and contractive invariant sets defined by a -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 -norm are equivalent for the considered class of matrix polytopes (fact which is not true for general matrix polytopes).
|M| denotes the matrix built with the absolute values of the entries of M. (xvii) M  ∈ R × ( superscript from Schur) denotes the nonnegative matrix defined by M  = |M|.(xviii) M  ∈ R × ( superscript from Hurwitz) denotes the essentially nonnegative matrix defined by M  = M  + |M  |, where M  = diag{ 11 , ...,   } and M  = M − M  .