SOME INVERSE RESULTS ABOUT MATRIX STABILITY IN TERMS OF MATRIX MEASURE

The aim of the present work is for each complex matrix A ∈ C with spectral abscissa S(A) < 0 and for each vector norm ||.|| in C to establish an explicit construction to obtain a matrix measure μ||.|| with μ||.||(A) < 0. Furthermore the proved result is extended for the case when the entries ajk of the matrix A are continuous functions, i.e. a j k ∈ C(K,C), where K ⊂ C̄+ is an arbitrary compact set. AMS Subject Classification: 15A60, 37C75, 65F35


Introduction
Many mathematical facts about matrix stability are playing an important role for numerous concrete applications as useful tools for practical calculation on the base of which we can make important qualitative conclusions.For example it is well known that if a square matrix A = {a j k } n k,j=1 ∈ C n×n is stable (i.e.all eigenvalues of have negative real parts) then the zero solution of the corresponding first order (or fractional order α ∈ (0, 1)) linear autonomous system of differential equations is globally asymptotical stable.An explicit suf-ficient condition which guaranties that the eigenvalues have negative real parts is µ(A) < 0, where µ : C n×n → R is an arbitrary matrix measure (also known as logarithmic norm or Losinskii measure).Unfortunately it is not difficult to create an example which shows that the inverse statement is generally not true, i.e. all eigenvalues of the matrix can have negative real parts but it is possible that there exists a matrix measure with µ(A) ≥ 0. This fact leads to great difficulties in the proofs of several results from hereditary and comparison type.
An inverse result is proved from Zahreddine in [10] (Theorem 2.11).The statement of this theorem is that S(A) = inf ||.||∈Υ µ ||.|| (A), where S(A) denotes the spectral abscissa of the matrix A, Υ denotes the set of all possible vector norms in C n and with µ ||.|| (A) is denoted the corresponding matrix measure induced from an arbitrary norm ||.|| ∈ Υ.This work was the first motivation to extend and improve the results obtained in it for practical application.The main consequence of the proved result in [10] is that if S(A) < 0 then there exists at least one matrix measure µ ||.|| with µ ||.|| (A) < 0. It is an elegant mathematical result, but from point of view of the applications it is very difficult (maybe impossible) to calculate the infimum in this relation over all possible vector norms in C n×n , which decrease essentially the practical application of this result.More detailed, this result cannot help us to find practically some matrix measure µ ||.|| with µ ||.|| (A) < 0 or for a concrete norm used in some proof to conclude that for the matrix measure µ ||.|| induced from this norm, the inequality µ ||.|| (A) < 0 holds.These are the main reasons for our concept for each matrix A ∈ C n×n with spectral abscissa S(A) < 0 and for each vector norm ||.|| in C n to introduce an explicit construction under which we obtain a matrix measure µ ||.|| with µ ||.|| (A) < 0.
The paper is organized as follows.In Section 2, we recall some needed definitions and some results about the matrix measure (named also logarithmic norm [1] or Lozinskii measure [5]) for square complex matrices.Section 3 is devoted to our main results about the matrix measure for complex matrices.We prove that for each complex matrix A ∈ C n×n with spectral abscissa S(A) < 0 and for each vector norm ||.|| in C n can be explicitly constructed a matrix measure µ ||.|| with µ ||.|| (A) < 0. Furthermore the proved result is extended for the case when the entries a j k of the matrix A are continuous functions, i.e. a j k ∈ C(k, C), where K ⊂ C+ is an arbitrary compact set.The last mentioned result can be used for example in the stability analysis of the neutral linear fractional differential systems even with distributed delays.
Note that our considerations are related with the stability analysis as well as with estimation of eigenvalue range for complex interval and parametric matrices.For more details we refer the works of O. Pastravanu at all [6,7,8], L. Kolev at all [2,3,4] and the references therein.

Preliminaries
In this section to avoid possible misunderstandings, we recall some definitions and results about the matrix measure (named also logarithmic norm or Lozinskii measure) for complex matrices.
Let denote the transpose, the conjugate and the Hermitian transpose of A respectively.With Sp(A) we denote the spectrum of A, ρ(A) = max |λ| is the spectral radii and S(A) = sup{Reλ|λ ∈ Sp(A)} the spectral abscissa of A.
Let denote with Υ the set of all possible vector norms in C n and let ||.|| ∈ Υ be an arbitrary norm.For H ∈ C n×n we define the induced matrix norm in ||Hz||.

Definition 1. ([1]
, [5]) For an arbitrary norm ||.|| ∈ Υ the corresponding matrix measure is defined by Note that if a matrix norm ||.|| is not induced by some vector norm in C n (as described above), then for the existence of the limit in Definition 1 the matrix norm must satisfy two additional conditions: ||I|| = 1 and ||AB|| ≤ ||A||||B|| for each A, B ∈ C n×n where I is the identity-matrix.
Using an arbitrary norm ||.|| ∈ Υ as generating norm, we define the corresponding family of weighted norms as follow: For every A, B ∈ C n×n , α ≥ 0, and p ∈ C the following relations hold: In the partial case for the Holder vector q-norm defined by ||z|| q = ( n i=1 , ||z|| ∞ = max 1≤i≤n {|z i |}, the corresponding matrix measure can be calculated explicitly in the cases : ) The logarithmic inefficiency of an arbitrary norm ||.|| ∈ Υ with respect to the matrix A is given by q(A) = µ(A) − S(A), where µ(A) is the corresponding induced matrix measure.

Definition 3. ([9]
) A norm and the corresponding matrix measure are logarithmically optimal with respect to A if q(A) = 0 and logarithmically ǫefficient if for some ǫ > 0 we have q(A) ≤ ǫ.

Main Results
The statement of the next theorem is a practically applicable generalization of Theorem 2.11 in [10].As a generating norm we will take an arbitrary norm ||.|| ∈ Υ, but in the practice as generating norms are used the mostly applicable norms as the Holder's vector q-norm, q ∈ N ∪ {∞}.
∈ C n×n be an arbitrary stable matrix, i.e.S(A) < 0 and ||.|| ∈ Υ be an arbitrary norm.Then according Jordan's theorem there exists a nonsingular matrix H ∈ C n×n , such that J = HAH −1 = ∆ + U , where ∆ is a diagonal matrix and U is a strictly upper triangular matrix, which offdiagonal entries are the same as the off-diagonal entries of J.For each δ > 0 and H ∈ C n×n with detH = 0 we introduce the matrix H δ as follows: Then denoting y = H δ z, for each η > 0 we have From (3.2) it follows Then for each δ ∈ (0, ǫ||U || −1 ) from (3.3) we obtain that holds for every ǫ > 0.
(ii) Since S(A) < 0 then the assertion (ii) follows from (3.4) for every ǫ ∈ (0, 2 −1 |S(A)|.Remark 6. Theorem 2.11 in [10] [10] is that we can take every practically used norm as generating norm and construct the corresponding family of weighted norms.Then we can take the infimum only over the constructed family of weighted norms (which obviously can be different for the different generating norms) instead of calculating infimum over all possible norms, which can be very difficult or maybe impossible.Theorem 4 and Corollary 5 provide several possibilities to obtain results concerning stability for example of fractional differential systems with delays.But as a shortcoming of the obtained results we can point its "locality".Usually the entries of the characteristic matrix of the (fractional) differential or functional-differential systems with delays are continuous (even entire) functions of complex variable A(p) = {a j k (p)} n k,j=1 , p ∈ C, i.e. in general the entries are not constants.Then there is no guarantee that if for some p * ∈ C the corresponding induced matrix measure µ H δ is logarithmically ǫ-efficient with respect to A(p * ), the same matrix measure will be logarithmically ǫ-efficient with respect to A(p) for some other p = p * .The next theorem is an important step to overcome this obstacle.Theorem 8. Let A, B ∈ C n×n be arbitrary stable matrices, ||.|| ∈ Υ be an arbitrary norm and ǫ > 0 be an arbitrary number.
Remark 9. Obviously the assertion of Theorem 8 can be extended for an arbitrary finite number l ∈ N of matrices.We proved only the case l = 2 to avoid unnecessarily additional writing.).The rest of the proof can be completed as the proof of Theorem 10.
Remark 13.It is clear that the assertion of Theorem 12 remains true for arbitrary finite number of matrices A i (p), i = 1, 2, . . ., l, l ∈ N.