| Title:
|
Asymptotic behaviour of solutions of real two-dimensional differential system with nonconstant delay (English) |
| Author:
|
Rebenda, Josef |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
45 |
| Issue:
|
3 |
| Year:
|
2009 |
| Pages:
|
223-236 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
stability |
| Keyword:
|
asymptotic behaviour |
| Keyword:
|
differential system |
| Keyword:
|
nonconstant delay |
| Keyword:
|
Lyapunov method |
| MSC:
|
34K12 |
| MSC:
|
34K20 |
| MSC:
|
34K25 |
| idZBL:
|
Zbl 1212.34235 |
| idMR:
|
MR2591678 |
| . |
| Date available:
|
2009-09-18T11:22:43Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134228 |
| . |
| Reference:
|
[1] Kalas, J.: Asymptotic behaviour of a two-dimensional differential systems with nonconstant delay.accepted in Math. Nachr. |
| Reference:
|
[2] Kalas, J., Baráková, L.: Stability and asymptotic behaviour of a two-dimensional differential system with delay.J. Math. Anal. Appl. 269 (2002), 278–300. Zbl 1008.34064, MR 1907886, 10.1016/S0022-247X(02)00023-9 |
| Reference:
|
[3] Ráb, M., Kalas, J.: Stability of dynamical systems in the plane.Differential Integral Equations 3 (1990), 124–144. MR 1014730 |
| Reference:
|
[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. Zbl 1157.34356, MR 2391945 |
| Reference:
|
[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. |
| Reference:
|
[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. Zbl 1169.34051, MR 2484509 |
| . |
