On the asymptotic stability of linear difference equations with time-varying coefficients

Vasilii Zaitsev; Vasilii Zaitsev

doi:10.3934/math.20231207

AIMS Mathematics

2023, Volume 8, Issue 10: 23734-23746. doi: 10.3934/math.20231207

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On the asymptotic stability of linear difference equations with time-varying coefficients

Vasilii Zaitsev ^,

Laboratory of Mathematical Control Theory, Udmurt State University, ul. Universitetskaya, 1, Izhevsk, 426034, Russia

Received: 18 April 2023 Revised: 23 July 2023 Accepted: 27 July 2023 Published: 02 August 2023
MSC : 39A06, 39A30

Issues of the asymptotic stability for linear difference equations with time-varying coefficients are discussed. It is shown that, in contrast to equations with constant coefficients, the condition of Schur stability of the characteristic polynomial for a linear difference equation with time-varying coefficients is neither necessary nor sufficient for the asymptotic stability of the difference equation. It is proved that the analog of Kharitonov's theorem on robust stability and the edge theorem do not hold for a difference equation if the coefficients of the equation are not constant.
- linear difference equations,
- asymptotic stability,
- time-varying system,
- Schur stability,
- robust stability
Citation: Vasilii Zaitsev. On the asymptotic stability of linear difference equations with time-varying coefficients[J]. AIMS Mathematics, 2023, 8(10): 23734-23746. doi: 10.3934/math.20231207

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Abstract

Issues of the asymptotic stability for linear difference equations with time-varying coefficients are discussed. It is shown that, in contrast to equations with constant coefficients, the condition of Schur stability of the characteristic polynomial for a linear difference equation with time-varying coefficients is neither necessary nor sufficient for the asymptotic stability of the difference equation. It is proved that the analog of Kharitonov's theorem on robust stability and the edge theorem do not hold for a difference equation if the coefficients of the equation are not constant.

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