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Astrovskii, A. I.; Gaishun, I. V.

Relationship between canonical forms of linear differential observation systems and canonical forms of their discrete approximations. (English. Russian original) Zbl 1227.93020

Differ. Equ. 47, No. 7, 963-971 (2011); translation from Differ. Uravn. 47, No. 7, 954-962 (2011).
Summary: We establish a relationship between the canonical form of a linear differential system and the canonical form of its discrete approximation based on the replacement of the derivative by Euler’s finite difference. We prove that if there exist limits of certain sequences of discrete functions constructed with the use of coefficients of the canonical form of the discrete system, then these limits define the canonical form of the differential system.

Cited in 3 Documents

MSC:

93B10 Canonical structure
93C15 Control/observation systems governed by ordinary differential equations
93C05 Linear systems in control theory

Keywords:

canonical form; linear differential system; discrete approximation; Euler’s finite difference
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References:

[1] Gaishun, I.V., Vvedenie v teoriyu lineinykh nestatsionarnykh sistem (Introduction to Theory of Linear Nonstationary Systems), Moscow, 2004.
[2] Astrovskii, A.I. and Gaishun, I.V., Quasidifferentiability and Observability of Linear Nonstationary Systems, Differ. Uravn., 2009, vol. 45, no. 11, pp. 1567–1576. · Zbl 1179.93045
[3] Astrovskii, A.I. and Gaishun, I.V., Quasidifferentiability and Canonical Forms of Linear Nonstationary Observation Systems, Differ. Uravn., 2010, vol. 46, no. 3, pp. 423–431. · Zbl 1191.93023
[4] Astrovskii, A.I. and Gaishun, I.V., One Way of Constructing Frobenius Canonical Forms of Linear Nonstationary Observation Systems, Differ. Uravn., 2010, vol. 46, no. 10, pp. 1479–1487. · Zbl 1210.93021
[5] Astrovskii, A.I. and Gaishun, I.V., Canonical Forms of Linear Nonstationary Observation Systems with Quasidifferentiable Coefficients under Various Transformation Groups, Differ. Uravn., 2011, vol. 47, no. 2, pp. 254–263. · Zbl 1227.93019 · doi:10.1134/S001226611102011X
[6] Gaishun, I.V., Sistemy s diskretnym vremenem (Systems with Discrete Time), Minsk, 2001.
[7] Derr, V.Ya., Nonoscillation of Solutions of a Linear Quasidifferential Equation, Izv. Inst. Mat. i Inform. Udmurt. Gos. Univ., Izhevsk, 1999, vol. 1(16), pp. 3–105.
[8] Teptin, A.L., On Certain Properties of Solutions of Linear Difference Equations Approximating Differential Equations in the Interval of Nonoscillation, Differ. Uravn., 1965, vol. 1, no. 4, pp. 478–498.
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