Criterion for the stability of the partial indices of a circular matrix function

1988 ◽  
Vol 44 (4) ◽  
pp. 775-780 ◽  
Author(s):  
P. M. Tishin
Keyword(s):  
Matrix Function ◽  
Partial Indices ◽  
The Stability

scholarly journals On effective criterion of stability of partial indices for matrix polynomials

2020 ◽  
Vol 476 (2238) ◽  
pp. 20200012
Author(s):  
N. V. Adukova ◽  
V. M. Adukov
Keyword(s):  
Small Perturbation ◽  
Stability Criterion ◽  
Matrix Function ◽  
Toeplitz Matrices ◽  
Matrix Functions ◽  
Matrix Polynomials ◽  
Partial Indices ◽  
The Stability ◽  
The Given

In the work, we obtain an effective criterion of the stability of the partial indices for matrix polynomials under an arbitrary sufficiently small perturbation. Verification of the stability is reduced to calculation of the ranks for two explicitly defined Toeplitz matrices. Furthermore, we define a notion of the stability of the partial indices in the given class of matrix functions. This means that we will consider an allowable small perturbation such that a perturbed matrix function belong to the same class as the original one. We prove that in the class of matrix polynomials the Gohberg–Krein–Bojarsky criterion is preserved, i.e. new stability cases do not arise. Our proof of the stability criterion in this class does not use the Gohberg–Krein–Bojarsky theorem.

scholarly journals TheZ-Transform Method and Delta Type Fractional Difference Operators

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Dorota Mozyrska ◽  
Małgorzata Wyrwas
Keyword(s):  
Matrix Function ◽  
Initial Value Problems ◽  
Stability Problem ◽  
Difference Operators ◽  
Fractional Order Systems ◽  
Initial Value ◽  
Fractional Difference ◽  
Transform Method ◽  
Delta Type ◽  
The Stability

The Caputo-, Riemann-Liouville-, and Grünwald-Letnikov-type difference initial value problems for linear fractional-order systems are discussed. We take under our consideration the possible solutions via the classicalZ-transform method. We stress the formula for the image of the discrete Mittag-Leffler matrix function in theZ-transform. We also prove forms of images in theZ-transform of the expressed fractional difference summation and operators. Additionally, the stability problem of the considered systems is studied.

scholarly journals On explicit Wiener–Hopf factorization of 2 × 2 matrices in a vicinity of a given matrix

2020 ◽  
Vol 476 (2238) ◽  
pp. 20200027 ◽  
Author(s):  
L. Ephremidze ◽  
I. Spitkovsky
Keyword(s):  
Numerical Simulations ◽  
Matrix Function ◽  
Factorization Method ◽  
Spectral Factorization ◽  
Engineering Applications ◽  
Partial Indices ◽  
Matrix Spectral Factorization

As it is known, the existence of the Wiener–Hopf factorization for a given matrix is a well-studied problem. Severe difficulties arise, however, when one needs to compute the factors approximately and obtain the partial indices. This problem is very important in various engineering applications and, therefore, remains to be subject of intensive investigations. In the present paper, we approximate a given matrix function and then explicitly factorize the approximation regardless of whether it has stable partial indices. For this reason, a technique developed in the Janashia–Lagvilava matrix spectral factorization method is applied. Numerical simulations illustrate our ideas in simple situations that demonstrate the potential of the method.

Partial Indices of Small Perturbations of a Degenerate Continuous Matrix Function

2002 ◽  
pp. 185-195
Author(s):  
Israel Feldman ◽  
Naum Krupnik ◽  
Alexander Markus
Keyword(s):  
Matrix Function ◽  
Small Perturbations ◽  
Partial Indices

The Lyapunov matrix function and the stability of hybrid systems

1985 ◽  
Vol 21 (4) ◽  
pp. 386-393
Author(s):  
A. A. Martynyuk
Keyword(s):  
Hybrid Systems ◽  
Matrix Function ◽  
Lyapunov Matrix ◽  
The Stability

scholarly journals Exact conditions for preservation of the partial indices of a perturbed triangular 2 × 2 matrix function

2020 ◽  
Vol 476 (2237) ◽  
pp. 20200099 ◽  
Author(s):  
Victor M. Adukov ◽  
Gennady Mishuris ◽  
Sergei V. Rogosin
Keyword(s):  
Unit Circle ◽  
Matrix Function ◽  
Triangular Matrix ◽  
Specific Class ◽  
Matrix Functions ◽  
Approximate Methods ◽  
Parametric Space ◽  
Partial Indices ◽  
Unstable Set ◽  
Factorization Technique

The possible instability of partial indices is one of the important constraints in the creation of approximate methods for the factorization of matrix functions. This paper is devoted to a study of a specific class of triangular matrix functions given on the unit circle with a stable and unstable set of partial indices. Exact conditions are derived that guarantee a preservation of the unstable set of partial indices during a perturbation of a matrix within the class. Thus, even in this probably simplest of cases, when the factorization technique is well developed, the structure of the parametric space (guiding the types of matrix perturbations) is non-trivial.

scholarly journals Regular approximate factorization of a class of matrix-function with an unstable set of partial indices

2018 ◽  
Vol 474 (2209) ◽  
pp. 20170279 ◽  
Author(s):  
G. Mishuris ◽  
S. Rogosin
Keyword(s):  
Matrix Function ◽  
Stable Set ◽  
Matrix Functions ◽  
Stable Sets ◽  
Approximate Factorization ◽  
Original Matrix ◽  
Partial Indices ◽  
Unstable Set ◽  
The Matrix ◽  
The Difference

From the classic work of Gohberg & Krein (1958 Uspekhi Mat. Nauk. XIII , 3–72. (Russian).), it is well known that the set of partial indices of a non-singular matrix function may change depending on the properties of the original matrix. More precisely, it was shown that if the difference between the largest and the smallest partial indices is larger than unity then, in any neighbourhood of the original matrix function, there exists another matrix function possessing a different set of partial indices. As a result, the factorization of matrix functions, being an extremely difficult process itself even in the case of the canonical factorization, remains unresolvable or even questionable in the case of a non-stable set of partial indices. Such a situation, in turn, has became an unavoidable obstacle to the application of the factorization technique. This paper sets out to answer a less ambitious question than that of effective factorizing matrix functions with non-stable sets of partial indices, and instead focuses on determining the conditions which, when having known factorization of the limiting matrix function, allow to construct another family of matrix functions with the same origin that preserves the non-stable partial indices and is close to the original set of the matrix functions.

On the stability of a system of two linear hybrid functional differential systems with aftereffect

2020 ◽  
pp. 299-306
Author(s):  
Pyotr M. Simonov
Keyword(s):  
Differential Equations ◽  
Matrix Function ◽  
Sufficient Conditions ◽  
Hybrid Functional ◽  
Cauchy Matrix ◽  
Linear Ordinary Differential Equations ◽  
Functional Differential Systems ◽  
Functional Differential ◽  
Linear Differential ◽  
The Stability

We consider a system of two hybrid vector equations containing linear difference (defined on a discrete set) and functional differential (defined on a half-axis) parts. To study it, a model system of two vector equations is chosen, one of which is linear difference with aftereffect (LDEA), and the other is a linear functional differential with aftereffect (LFDEA). Two equivalent representations of this system are shown: the first representation in the form of LFDEA, the second — in the form of LDEA. This allows us to study the stability issues of the system under consideration using the well-known results on the stability of LFDEA and LDEA. Using the results of the article [Gusarenko S. A. On the stability of a system of two linear differential equations with delayed argument // Boundary value problems. Interuniversity collection of scientific papers. Perm: PPI, 1989. P. 3–9], two examples are shown when a joint system of four equations will be stable with respect to the right side. In the first example, we use the LFDEA for which sufficient conditions for the sign-definiteness of the elements of the 2 Ч 2 Cauchy matrix function are known (in terms of the LFDEA coefficients). In the second example, LFDEA is given such that LFDEA is a system of linear ordinary differential equations (LODE) of the second order. In both cases, estimates of the components of the Cauchy matrix function are known. An exponential estimate with a negative exponent is given for the components of the Cauchy matrix function of LDEA.

Open discussion

1982 ◽  
Vol 99 ◽  
pp. 605-613
Author(s):  
P. S. Conti
Keyword(s):  
Buenos Aires ◽  
Large Mass ◽  
List Type ◽  
Common Phenomenon ◽  
Open Discussion ◽  
The Stability ◽  
Ring Nebulae

Conti: One of the main conclusions of the Wolf-Rayet symposium in Buenos Aires was that Wolf-Rayet stars are evolutionary products of massive objects. Some questions:–Do hot helium-rich stars, that are not Wolf-Rayet stars, exist?–What about the stability of helium rich stars of large mass? We know a helium rich star of ∼40 MO. Has the stability something to do with the wind?–Ring nebulae and bubbles : this seems to be a much more common phenomenon than we thought of some years age.–What is the origin of the subtypes? This is important to find a possible matching of scenarios to subtypes.

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