IJPAM: Volume 111, No. 3 (2016)

Title

D-DECOMPOSITION METHOD FOR STABILITY CHECKING
FOR TRINOMIAL LINEAR DIFFERENCE EQUATION
WITH TWO DELAYS

Authors

Mikhai Kipnis$^1$, Ravil Nigmatulin$^2$
$^{1,2}$Department of Mathematics and Physics
South Ural State Pedagogical University for the Humanities
69 Lenin Avenue, Chelyabinsk, 454080, RUSSIA

Abstract

We give asymptotic stability boundaries in the parameter space of trinomial linear difference equation, and also we give explicit inequalities for stability checking of the equation. We study a generalization of the notion of stability which we call r-stability.

History

Received: September 10, 2016
Revised: November 2, 2016
Published: December 19, 2016

AMS Classification, Key Words

AMS Subject Classification: 39A30
Key Words and Phrases: asymptotic stability, difference equation, two delays, D-decomposition method

Download Section

Download paper from here.
You will need Adobe Acrobat reader. For more information and free download of the reader, see the Adobe Acrobat website.

Bibliography

1
S. Levin, R. May, A note on difference-delay equations, Theoret. Popul. Biol., 9 (1976), 178-187.

2
S.A. Kuruklis, The asymptotic stability of $x_{n+1}-a x_n+ b x_{n-k}=0$, J. Math. Anal. Appl., 188 (1994), 719-731, doi: https://doi.org/10.1006/jmaa.1994.1457.

3
Yu. P. Nikolaev, The set of stable polynomials of discrete systems: its geometry, Autom. Remote Control, 63, No. 7 (2002), 1080-1088, doi: https://doi.org/10.1023/A:1016154714222.

4
F. Dannan, The asymptotic stability of $x(n+k)+ax(n)+bx(n-l)=0$, J. Difference Equ. Appl., 7, No. 6 (2004), 1-11, doi: https://doi.org/10.1080/10236190410001685058.

5
M. Kipnis, R. Nigmatulin, Stability of the trinomial linear difference equations with two delays, Autom. Remote Control, 65, No. 11 (2004), 1710-1723, doi: https://doi.org/10.1023/B:AURC.0000047886.46498.79.

6
S.A. Ivanov, M.M. Kipnis, V.V. Malygina, The stability cone for a difference matrix equation with two delays, ISRN Applied Math., 2011 (2011), 1-19, doi: https://doi.org/10.5402/2011/910936.

7
S.S.Cheng, S.Y. Huang, Alternate derivations of the stability region of a difference equation with two delays, Appl. Math. E-Notes, 9 (2009), 225-253.

8
J. Čermák, J. Jánský, Explicit stability conditions for a linear trinomial delay difference equation, Appl. Math. Letters, 43 (2015), 56-60, doi: https://doi.org/10.1016/j.aml.2014.11.014.

9
E.N. Gryazina, The D-Decomposition theory, Autom. Remote Control, 65, No. 12 (2004), 1872-1884, doi: https://doi.org/10.1023/B:AURC.0000049874.93222.2c.

10
E.N. Gryazina, B.T. Polyak, Stability regions in the parameter space: D-decomposition revisited, Automatica, 42, No. 1 (2006), 13-26, doi: https://doi.org/10.1016/j.automatica.2005.08.010.

11
Yu.I. Neimark, Finding the parameter values that make automatic control system stable, Autom. Remote Control, 9, No. 3 (1948), 190-203 (in Russian).

How to Cite?

DOI: 10.12732/ijpam.v111i3.11 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2016
Volume: 111
Issue: 3
Pages: 479 - 489


Google Scholar; DOI (International DOI Foundation); WorldCAT.

CC BY This work is licensed under the Creative Commons Attribution International License (CC BY).