The method of Lyapunov functions for systems of linear difference equations with almost periodic coefficients

Kirichenova, O. V.; Kotyurgina, A. S.; Romanovskii, R. K.

doi:10.1007/bf02104765

Cited by 12 publications

(3 citation statements)

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“…The justification rests on the properties of some functions on the compact set K = H × S, where H is the hull of A(t), and S is the unit sphere in C N . Later the results of [1] were extended to other classes of almost periodic equations: difference, difference-differential, and functional-differential equations [2][3][4][5][6][7][8][9].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Exponential dichotomy of linear systems with almost periodic matrices

Romanovskii

Bel’gart

2009

Sib Math J

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Using the Lyapunov direct method, we establish a condition of exponential dichotomy for the class of dynamical systems under weaker assumptions as compared to the case of an arbitrary continuous matrix on the derivative of the Lyapunov function along the trajectories. By way of application, we obtain a sufficient condition for the dichotomy of a second order almost periodic vector equation in terms of coefficients.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Exponential dichotomy of linear systems with almost periodic matrices

Romanovskii

Bel’gart

2009

Sib Math J

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“…In [2,3] it was noted that, with some natural changes in the statement, this result extends to the nonautonomous systemṡ x = f (x, t) provided that the right-hand side and the Lyapunov function ν(x, t) are almost periodic in t (in the case linear systems, the talk is about exponential stability). Later in [4,5], these results were extended to the systems of difference equations…”

mentioning

confidence: 93%

“…In [7][8][9][10], where the results of [2,3] were extended to functional-differential equations of delay and neutral types, the lack of compactness of the unit sphere in the phase space was compensated by the compactness of trajectories. The question has been open until recently about extending the results of [2][3][4][5] to the case in which the phase space of a dynamical system is not locally compact.…”

mentioning

confidence: 99%