Asymptotic behaviour of solutions of real two-dimensional differential system with nonconstant delay

Josef Rebenda

Archivum Mathematicum (2009)

  • Volume: 045, Issue: 3, page 223-236
  • ISSN: 0044-8753

Abstract

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In this article, stability and asymptotic properties of solutions of a real two-dimensional system x ' ( t ) = 𝐀 ( t ) x ( t ) + 𝐁 ( t ) x ( τ ( t ) ) + 𝐡 ( t , x ( t ) , x ( τ ( t ) ) ) are studied, where 𝐀 , 𝐁 are matrix functions, 𝐡 is a vector function and τ ( t ) ≤ t is a nonconstant delay which is absolutely continuous and satisfies lim t → ∞ τ ( t ) = ∞ . Generalization of results on stability of a two-dimensional differential system with one constant delay is obtained using the methods of complexification and Lyapunov-Krasovskii functional and some new corollaries and examples are presented.

How to cite

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Rebenda, Josef. "Asymptotic behaviour of solutions of real two-dimensional differential system with nonconstant delay." Archivum Mathematicum 045.3 (2009): 223-236. <http://eudml.org/doc/250553>.

@article{Rebenda2009,
abstract = {In this article, stability and asymptotic properties of solutions of a real two-dimensional system $x^\{\prime \}(t) = \mathbf \{A\} (t) x(t) + \mathbf \{B\} (t) x (\tau (t)) + \mathbf \{h\} (t, x(t), x(\tau (t)))$ are studied, where $\mathbf \{A\}$, $\mathbf \{B\}$ are matrix functions, $\mathbf \{h\}$ is a vector function and $\tau (t) \le t$ is a nonconstant delay which is absolutely continuous and satisfies $\lim \limits _\{t \rightarrow \infty \} \tau (t) = \infty $. Generalization of results on stability of a two-dimensional differential system with one constant delay is obtained using the methods of complexification and Lyapunov-Krasovskii functional and some new corollaries and examples are presented.},
author = {Rebenda, Josef},
journal = {Archivum Mathematicum},
keywords = {stability; asymptotic behaviour; differential system; nonconstant delay; Lyapunov method; stability; asymptotic behaviour; differential system; nonconstant delay; Lyapunov method},
language = {eng},
number = {3},
pages = {223-236},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Asymptotic behaviour of solutions of real two-dimensional differential system with nonconstant delay},
url = {http://eudml.org/doc/250553},
volume = {045},
year = {2009},
}

TY - JOUR
AU - Rebenda, Josef
TI - Asymptotic behaviour of solutions of real two-dimensional differential system with nonconstant delay
JO - Archivum Mathematicum
PY - 2009
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 045
IS - 3
SP - 223
EP - 236
AB - In this article, stability and asymptotic properties of solutions of a real two-dimensional system $x^{\prime }(t) = \mathbf {A} (t) x(t) + \mathbf {B} (t) x (\tau (t)) + \mathbf {h} (t, x(t), x(\tau (t)))$ are studied, where $\mathbf {A}$, $\mathbf {B}$ are matrix functions, $\mathbf {h}$ is a vector function and $\tau (t) \le t$ is a nonconstant delay which is absolutely continuous and satisfies $\lim \limits _{t \rightarrow \infty } \tau (t) = \infty $. Generalization of results on stability of a two-dimensional differential system with one constant delay is obtained using the methods of complexification and Lyapunov-Krasovskii functional and some new corollaries and examples are presented.
LA - eng
KW - stability; asymptotic behaviour; differential system; nonconstant delay; Lyapunov method; stability; asymptotic behaviour; differential system; nonconstant delay; Lyapunov method
UR - http://eudml.org/doc/250553
ER -

References

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  1. Kalas, J., Asymptotic behaviour of a two-dimensional differential systems with nonconstant delay, accepted in Math. Nachr. 
  2. Kalas, J., Baráková, L., 10.1016/S0022-247X(02)00023-9, J. Math. Anal. Appl. 269 (2002), 278–300. (2002) Zbl1008.34064MR1907886DOI10.1016/S0022-247X(02)00023-9
  3. Ráb, M., Kalas, J., Stability of dynamical systems in the plane, Differential Integral Equations 3 (1990), 124–144. (1990) MR1014730
  4. Rebenda, J., Asymptotic properties of solutions of real two-dimensional differential systems with a finite number of constant delays, Mem. Differential Equations Math. Phys. 41 (2007), 97–114. (2007) Zbl1157.34356MR2391945
  5. Rebenda, J., Stability of the trivial solution of real two-dimensional differential system with nonconstant delay, In 6. matematický workshop - sborník, FAST VUT Brno 2007, 2007, 49–50 (abstract). Fulltext available at http://math.fce.vutbr.cz/~pribyl/workshop_2007/prispevky/Rebenda.pdf. (2007, 49–50 (abstract). Fulltext available at http://math.fce.vutbr.cz/~pribyl/workshop_2007/prispevky/Rebenda.pdf) 
  6. Rebenda, J., Asymptotic behaviour of real two-dimensional differential system with a finite number of constant delays, Demonstratio Math. 41 (4) (2008), 845–857. (2008) Zbl1169.34051MR2484509

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