Non-delayed Linear Planar Discrete Systems with Constant Coefficients Equivalent to Linear Planar Discrete Delayed Systems with Constant Coefficients

Non-delayed Linear Planar Discrete Systems with Constant Coefficients Equivalent to Linear Planar Discrete Delayed Systems with Constant Coefficients

Paper in proceedings outside WoS and Scopus

Planar linear discrete delayed systems w(k +1) = Aw(k)+Bw(k −m), k = 0,1,... are considered, where A and B are constant matrices and m is a positive integer denoting a delay. Recently, formulas have been derived for the general solution of such systems provided that one eigenvalue of A is nonzero while the other being zero, and that matrices A and B satisfy the conditions known for so-called weakly delayed systems. Conditions characterizing weakly delayed systems imply that there are only two roots of the relevant quasi-characteristic equation. The present paper is concerned with the problem of a possible replacement of the delayed systems in question with non-delayed planar ones x(k +1) = Cx(k), k = 0,1,... having an identical general solution. It is shown that this is possible if a transient interval is passed and possible forms of the matrix C are found. Some relationships with previously known results are commented.

Planar linear discrete delayed systems w(k +1) = Aw(k)+Bw(k −m), k = 0,1,... are considered, where A and B are constant matrices and m is a positive integer denoting a delay. Recently, formulas have been derived for the general solution of such systems provided that one eigenvalue of A is nonzero while the other being zero, and that matrices A and B satisfy the conditions known for so-called weakly delayed systems. Conditions characterizing weakly delayed systems imply that there are only two roots of the relevant quasi-characteristic equation. The present paper is concerned with the problem of a possible replacement of the delayed systems in question with non-delayed planar ones x(k +1) = Cx(k), k = 0,1,... having an identical general solution. It is shown that this is possible if a transient interval is passed and possible forms of the matrix C are found. Some relationships with previously known results are commented.

planar delayed system; weak delay; initial problem; equivalent discrete system; general solution

planar delayed system; weak delay; initial problem; equivalent discrete system; general solution

DIBLÍK, J.; HARTMANOVÁ, M.

2025

20.06.2024

Univerzita Obrany

Brno

978-80-7582-493-6

Matematika, informační technologie a aplikované vědy, MITAV 2024

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https://mitav.unob.cz/data/final%20Program%20MITAV%202024.pdf