On the regularity of discrete linear systems

Czornik, Adam; Nawrat, Aleksander

doi:10.1016/j.laa.2009.12.018

Cited by 20 publications

(13 citation statements)

References 3 publications

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“…Namely, let

where

, and define

where the infimum is taken over all bases

v_{1}, \dots, v_{q}

R^{q}

. It follows from results in , , , that

0 \leq π \leq γ \leq σ \leq q π .

The advantage of having various regularity coefficients instead of a single one is that in each specific situation it is often easier to compute or at least to estimate a specific one. This causes that it is of interest to understand whether some estimates involving each of them, such as in the nonuniform hyperbolicity theory, are the best possible.…”

Section: Introductionmentioning

confidence: 99%

Relations between regularity coefficients

Barreira

Valls

2016

Math. Nachr.

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For a bounded sequence of matrices defining a nonautonomous dynamics with discrete time, we obtain all possible relations between the regularity coefficients introduced by Lyapunov, Perron and Grobman. This includes considering general inequalities between the coefficients and showing that these inequalities are the best possible, in the sense that for any three nonnegative numbers satisfying them, and for no others, there exists a bounded sequence of matrices having the numbers respectively as Lyapunov, Perron and Grobman coefficients. Moreover, we establish inequalities between the three coefficients and some other regularity coefficients.

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“…Namely, let

where

, and define

where the infimum is taken over all bases

v_{1}, \dots, v_{q}

R^{q}

. It follows from results in , , , that

0 \leq π \leq γ \leq σ \leq q π .

Section: Introductionmentioning

confidence: 99%

Relations between regularity coefficients

Barreira

Valls

2016

Math. Nachr.

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“…There are several approaches to defining the exponential growth rate, all of which turn out to be equivalent. A trajectory based definition considers Lyapunov exponents [36][37][38][39][40][41][42][43][44][45][46][47][48] of individual trajectories, which are defined in the discrete-time case as…”

Section: Lyapunov Exponents Methodsmentioning

confidence: 99%

“…The quantity ln ρ(d) is known in the control theory literature as the greatest Lyapunov exponent. Properties of Lyapunov exponents were studied in [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51], where sufficient conditions so that Lyapunov exponents characterizes exponential stability for switched systems are shown. In particular, it was shown that the system (1) is absolutely stable if and only if sup…”

Section: Lyapunov Exponents Methodsmentioning

confidence: 99%

Stability and controllability of switched systems

Klamka¹,

Czornik²,

Niezabitowski³

2013

Bulletin of the Polish Academy of Sciences: Technical Sciences

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Abstract. The study of properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. This paper aims to briefly survey recent results on stability and controllability of switched linear systems. First, the stability analysis for switched systems is reviewed. We focus on the stability analysis for switched linear systems under arbitrary switching, and we highlight necessary and sufficient conditions for asymptotic stability. After that, we review the controllability results.

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“…Perron exponents, which are closely related to them, appeared for the first time in [61] and [62]. Both of them were introduced in the context of linear time-varying differential equations and since then, they are an object of intensive research both for continuous-time and discrete-time, have many generalizations and variants (see [1][2][3][4][5][6][9][10][11][12][16][17][18]20,21,[23][24][25][26][27][28][29][30][31][32][33][34][35][36][40][41][42][43][44][45]49,52,53,69]). Moreover, the problem of sensitivity to parameter variation has been widely investigated (see for example [31] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

On the number of upper Bohl exponents for diagonal discrete time-varying linear system

Babiarz

Czornik

Niezabitowski

2015

Journal of Mathematical Analysis and Applications

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The Bohl exponents, similarly as the Lyapunov exponents, are one of the most important numerical characteristics of dynamical systems used in control theory. By contrast with properties of the Lyapunov characteristics, properties of the Bohl exponents are much less investigated. In this paper we present some new properties of the upper Bohl exponents of the diagonal discrete time-varying linear system.

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On the regularity of discrete linear systems

Cited by 20 publications

References 3 publications

Relations between regularity coefficients

Relations between regularity coefficients

Stability and controllability of switched systems

On the number of upper Bohl exponents for diagonal discrete time-varying linear system

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