| Title:
|
Boundary layer phenomenon for three -point boundary value problem for the nonlinear singularly perturbed systems (English) |
| Author:
|
Vrabel, Robert |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
47 |
| Issue:
|
4 |
| Year:
|
2011 |
| Pages:
|
644-652 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
This paper deals with the three-point boundary value problem for the nonlinear singularly perturbed second-order systems. Especially, we focus on an analysis of the solutions in the right endpoint of considered interval from an appearance of the boundary layer point of view. We use the method of lower and upper solutions combined with analysis of the integral equation associated with the class of nonlinear systems considered here. (English) |
| Keyword:
|
singularly perturbed systems |
| Keyword:
|
three--point boundary value problem |
| Keyword:
|
method of lower and upper solutions |
| Keyword:
|
controller |
| MSC:
|
34A34 |
| MSC:
|
34A40 |
| MSC:
|
34B10 |
| MSC:
|
34E15 |
| MSC:
|
93C10 |
| idZBL:
|
Zbl 1227.93080 |
| idMR:
|
MR2884866 |
| . |
| Date available:
|
2011-09-23T11:31:22Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141664 |
| . |
| Reference:
|
[1] Artstein, Z.: Stability in the presence of singular perturbations.Nonlinear Anal. 34(6) (1998), 817–827. Zbl 0948.34029, MR 1636588, 10.1016/S0362-546X(97)00574-9 |
| Reference:
|
[2] Artstein, Z.: Singularly perturbed ordinary differential equations with nonautonomous fast dynamics.J. Dynam. Different. Eqs. 11 (1999), 297–318. Zbl 0937.34044, MR 1695247, 10.1023/A:1021981430215 |
| Reference:
|
[3] Artstein, Z., Gaitsgory, V.: The value function of singularly perturbed control system.Appl. Math. Optim. 41 (2000), 425–445. MR 1739395, 10.1007/s002459911022 |
| Reference:
|
[4] Gaitsgory, V.: On a pepresentation of the limit occupational measures set of a control system with applications to singularly perturbed control systems.SIAM J. Control Optim. 43(1) (2004), 325–340. MR 2082704, 10.1137/S0363012903424186 |
| Reference:
|
[5] Burman, E., Guzmán, J., Leykekhman, D.: Weighted error estimates of the continuous interior penalty method for singularly perturbed problems.IMA J. Numer. Anal. 29(2) (2009), 284–314. Zbl 1166.65054, MR 2491428, 10.1093/imanum/drn001 |
| Reference:
|
[6] Gaitsgory, V., Nguyen, M. T.: Multiscale singularly perturbed control systems: Limit occupational measures sets and averaging.SIAM J. Control Optim. 41(3) (2002), 954–974. Zbl 1027.34071, MR 1939878, 10.1137/S0363012901393055 |
| Reference:
|
[7] Gopal, M.: Modern Control System Theory.New Age International, New Delhi 1993. |
| Reference:
|
[8] Guo, Y., Ge, W.: Positive solutions for three-point boundary value problems with dependence on the first order derivative.J. Math. Anal. Appl. 290(1) (2004), 291–301. Zbl 1054.34025, MR 2032241, 10.1016/j.jmaa.2003.09.061 |
| Reference:
|
[9] Chang, K. W., Howes, F. A.: Nonlinear Singular Perturbation Phenomena: Theory and Applications.Springer-Verlag, New York 1984. Zbl 0559.34013, MR 0764395 |
| Reference:
|
[10] Khan, A., Khan, I., Aziz, T., Stojanovic, M.: A variable-mesh approximation method for singularly perturbed boundary-value problems using cubic spline in tension.Internat. J. Comput. Math. 81 (2004), 12, 1513–1518. Zbl 1064.65066, MR 2169101, 10.1080/00207160412331284169 |
| Reference:
|
[11] Khan, R. A.: Positive solutions of four-point singular boundary value problems.Appl. Math. Comput. 201 (2008), 762–773. Zbl 1152.34016, MR 2431973, 10.1016/j.amc.2008.01.014 |
| Reference:
|
[12] Kokotovic, P., Khali, H. K., O’Reilly, J.: Singular Perturbation Methods in Control, Analysis and Design.Academic Press, London 1986. MR 0937051 |
| Reference:
|
[13] Natesan, S., Ramanujam, M.: Initial-value technique for singularly-perturbed turning-point problems exhibiting twin boundary layers.J. Optim. Theory Appl. 99 (1998), 1, 37–52. Zbl 0983.34050, MR 1653241, 10.1023/A:1021744025980 |
| Reference:
|
[14] Vrabel, R.: Three point boundary value problem for singularly perturbed semilinear differential equations.E. J. Qualitative Theory of Diff. Equ. 70 (2009), 1–4. Zbl 1195.34030, MR 2577505 |
| . |
