Asymptotic behaviour of a two-dimensional differential system with non-constant delay

The asymptotic behaviour and stability properties are studied for a real two‐dimensional system x′(t) = A(t)x (t) + B(t)x (θ (t)) + h (t, x (t), x (θ (t))), with a nonconstant delay t ‐ θ (t) ≥ 0. It is supposed that A,B and h are matrix functions and a vector function, respectively. The method of i...

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Bibliographic Details
Published in:Mathematische Nachrichten 2010-06, Vol.283 (6), p.879-890
Main Author: Kalas, Josef
Format: Article
Language:English
Subjects:
asymptotic behaviour
boundedness of solutions
Delayed differential system
stability
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Summary:The asymptotic behaviour and stability properties are studied for a real two‐dimensional system x′(t) = A(t)x (t) + B(t)x (θ (t)) + h (t, x (t), x (θ (t))), with a nonconstant delay t ‐ θ (t) ≥ 0. It is supposed that A,B and h are matrix functions and a vector function, respectively. The method of investigation is based on the transformation of the considered real system to one equation with complex‐valued coefficients. Stability and asymptotic properties of this equation are studied by means of a suitable Lyapunov‐Krasovskii functional. The results generalize the great part of the results of J. Kalas and L. Baráková [J. Math. Anal. Appl. 269, No. 1, 278–300 (2002)] for two‐dimensional systems with a constant delay (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
ISSN:0025-584X
1522-2616
DOI:10.1002/mana.200710026