On the D-stability and additive D-stability of matrices and Svicobians

Kosov, A. A.

doi:10.1134/s1990478910020092

Cited by 4 publications

(13 citation statements)

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“…In [169], Kosov proved the inclusion relations between diagonally stable, partially stable and P 0 -matrices, as well as properties of principal submatrices of multiplicative (additive) D-stable matrices first proved by Cross (see [80]), using the transition to the corresponding systems of differential equations. The author not only disproved the conjecture of [253], but also showed that a stronger condition of a matrix to be partially stable is also not sufficient for additive D-stability.…”

Section: -Kaszkurewicz Bhayamentioning

confidence: 99%

“…This concept fist appeared in a remark in [138] by Johnson under the name of "partial D-stability". However, the results based on this concept appeared only recently (see [169], where this concept was independently introduced).…”

Section: Multiplicative (D G)-stabilitymentioning

confidence: 99%

“…Additive interval D-stability (Romanishin and Sinitskii, 2002) For studying additive D-stability, the following subset of M n×n is considered (see [169], [218]):…”

Section: Additive (D G)-stabilitymentioning

confidence: 99%

“…Theorem 67 [169] For a P 0 -matrix A to be D-stable with respect to Θ, it suffices the four Kharitonov polynomials corresponding to F (λ), be stable.…”

Section: Analysis Of the Solvability Of Lyapunov Equation (8)mentioning

confidence: 99%

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Unifying Matrix Stability Concepts with a View to Applications

Kushel¹

2019

SIAM Rev.

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Multiplicative and additive D-stability, diagonal stability, Schur Dstability, H-stability are classical concepts which arise in studying linear dynamical systems. We unify these types of stability, as well as many others, in one concept of (D, G, •)-stability, which depends on a stability region D ⊂ C, a matrix class G and a binary matrix operation •. This approach allows us to unite several well-known matrix problems and to consider common methods of their analysis. In order to collect these methods, we make a historical review, concentrating on diagonal and D-stability. We prove some elementary properties of (D, G, •)-stable matrices, uniting the facts that are common for many partial cases. Basing on the properties of a stability region D which may be chosen to be a concrete subset of C (e.g. the unit disk) or to belong to a specified type of regions (e.g. LMI regions) we briefly describe the methods of further development of the theory of (D, G, •)-stability. We mention some applications of the theory of (D, G, •)-stability to the dynamical systems of different types.

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Section: -Kaszkurewicz Bhayamentioning

confidence: 99%

Section: Multiplicative (D G)-stabilitymentioning

confidence: 99%

“…Additive interval D-stability (Romanishin and Sinitskii, 2002) For studying additive D-stability, the following subset of M n×n is considered (see [169], [218]):…”

Section: Additive (D G)-stabilitymentioning

confidence: 99%

“…Theorem 67 [169] For a P 0 -matrix A to be D-stable with respect to Θ, it suffices the four Kharitonov polynomials corresponding to F (λ), be stable.…”

Section: Analysis Of the Solvability Of Lyapunov Equation (8)mentioning

confidence: 99%

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Unifying Matrix Stability Concepts with a View to Applications

Kushel¹

2019

SIAM Rev.

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“…Among the approaches to D-stability study, besides small-dimensions analysis, we should mention qualitative stability analysis (see [94], [138]), different determinantal conditions ( [61], [94], [44], [117]), generalizations of diagonally dominant and M -matrices ( [113]), study of matrix scalings ( [44]), diagonal stability condition ( [9], [94], [55]), studying Hadamard products, different matrix subclasses and special forms ( [94], [57], [46]), generalized singular value approach ( [97], [50], [121]). Verifiability of the existing conditions is widely discussed, new criteria and approaches appear (by spectral radius minimization in [109], structured singular value approach in [50], [121], by Kharitonov criterion in [115]). A method of checking D-stability by solving a number of LMI was proposed in [70].…”

mentioning

confidence: 99%

On a criterion of D-stability for P-matrices

Kushel

2016

Special Matrices

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The Problem of Generalized D-Stability in Unbounded LMI Regions and Its Computational Aspects

Kushel

Pavani

2020

J Dyn Diff Equat

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We generalize the concepts of D-stability and additive D-stability of matrices. For this, we consider a family of unbounded regions defined in terms of Linear Matrix Inequalities (so-called LMI regions). We study the problem when the localization of a matrix spectrum in an unbounded LMI region is preserved under specific multiplicative and additive perturbations of the initial matrix. The most well-known particular cases of unbounded LMI regions (namely, conic sectors and shifted halfplanes) are considered. A new Dstability criterion as well as sufficient conditions for generalized D-stability are analyzed. Several applications of the developed theory to dynamical systems are shown.

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On the D-stability and additive D-stability of matrices and Svicobians

Cited by 4 publications

References 2 publications

Unifying Matrix Stability Concepts with a View to Applications

Unifying Matrix Stability Concepts with a View to Applications

On a criterion of D-stability for P-matrices

The Problem of Generalized D-Stability in Unbounded LMI Regions and Its Computational Aspects

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