Stability of matrices with sufficiently strong negative-dominant-diagonal submatrices

Nieuwenhuis, Herman J.; Schoonbeek, Lambert

doi:10.1016/s0024-3795(96)00193-0

Cited by 10 publications

(14 citation statements)

References 6 publications

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“…In addition, based on Sylvester's theorem [25][26][27], the number of negative diagonal elements of matrix ( 5) is equal to the number of negative eigenvalues of this matrix. Since all diagonal elements (5), taking into account ( 2) and (3), are negative, all roots of the characteristic equation have negative real parts.…”

Section: Analysis Of the Characteristic Polynomial Rootsmentioning

confidence: 99%

Transient Behavior of the MAP/M/1/N Queuing System

2021

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This paper investigates the characteristics of the MAP/M/1/N queuing system in the transient mode. The matrix method for solving the Kolmogorov equations is proposed. This method makes it possible, in general, to obtain the main characteristics of the considered queuing system in a non-stationary mode: the probability of losses, the time of the transient mode, the throughput, and the number of customers in the system at time t. The developed method is illustrated by numerical calculations of the characteristics of the MAP/M/1/3 system in the transient mode.

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Section: Analysis Of the Characteristic Polynomial Rootsmentioning

confidence: 99%

Transient Behavior of the MAP/M/1/N Queuing System

2021

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“…In this paper, we generalize the concept of diagonal dominance. First, let us recall the classical definition of diagonally dominant matrices (see, for example, [14], also [27], [28]). Definition 1.…”

Section: Diagonal Dominance and Matrix Stabilitymentioning

confidence: 99%

“…The concept of negative diagonal dominance is also applied to establishing the stability of systems of second-order differential equations (see [27], [28]), with the applications to the stability of mechanical systems and certain economic models.…”

Section: Diagonal Dominance and Matrix Stabilitymentioning

confidence: 99%

“…Following [28], we re-write System (11) formally as the system of first-order differential equations:…”

Section: Stability Of Second Order Systemsmentioning

confidence: 99%

“…Going into details, System ( 11) is asymptotically stable if and only if the matrix A is stable, i.e. all its eigenvalues have negative real parts (see, for example, [27], [28]). Since the matrix A obviously does not preserve the structural properties of A and B, such as diagonal dominance, we will use the following lemma which describes the spectrum of A (see [27], p. 187, Lemma 2, Part (a)).…”

Section: Stability Of Second Order Systemsmentioning

confidence: 99%

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Generalized D-stability and diagonal dominance with applications to stability and transient response properties of systems of ODE

Kushel,

Pavani

2021

Preprint

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In this paper, we introduce the class of diagonally dominant (with respect to a given LMI region D ⊂ C) matrices that possesses the analogues of well-known properties of (classical) diagonally dominant matrices, e.g their spectra are localized inside the region D. Moreover, we show that in some cases, diagonal D-dominance implies (D, D)-stability ( i.e. the preservation of matrix spectra localization under multiplication by a positive diagonal matrix). Basing on the properties of diagonal stability and diagonal dominance, we analyze the conditions for stability of second-order dynamical systems. We show that these conditions are preserved under system perturbations of a specific form (so-called D-stability). We apply the concept of diagonal Ddominance to the analysis of the minimal decay rate of second-order systems and its persistence under specific perturbations (so-called relative D-stability). Diagonal D-dominance with respect to some conic region D is also shown to be a sufficient condition for stability and D-stability of fractional-order systems.

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Novel Versions of D-Stability in Matrices Provide New Insights into ODE Dynamics

Kushel

Pavani

2023

Mediterr. J. Math.

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Stability of matrices with sufficiently strong negative-dominant-diagonal submatrices

Cited by 10 publications

References 6 publications

Transient Behavior of the MAP/M/1/N Queuing System

Transient Behavior of the MAP/M/1/N Queuing System

Generalized D-stability and diagonal dominance with applications to stability and transient response properties of systems of ODE

Novel Versions of D-Stability in Matrices Provide New Insights into ODE Dynamics

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