Novel method for generalized stability analysis of nonlinear impulsive evolution equations

JinRong Wang; Yong Zhou; Wei Wei

Kybernetika (2012)

  • Volume: 48, Issue: 6, page 1211-1228
  • ISSN: 0023-5954

Abstract

top
In this paper, we discuss some generalized stability of solutions to a class of nonlinear impulsive evolution equations in the certain piecewise essentially bounded functions space. Firstly, stabilization of solutions to nonlinear impulsive evolution equations are studied by means of fixed point methods at an appropriate decay rate. Secondly, stable manifolds for the associated singular perturbation problems with impulses are compared with each other. Finally, an example on initial boundary value problem for impulsive parabolic equations is illustrated to our theory results.

How to cite

top

Wang, JinRong, Zhou, Yong, and Wei, Wei. "Novel method for generalized stability analysis of nonlinear impulsive evolution equations." Kybernetika 48.6 (2012): 1211-1228. <http://eudml.org/doc/251392>.

@article{Wang2012,
abstract = {In this paper, we discuss some generalized stability of solutions to a class of nonlinear impulsive evolution equations in the certain piecewise essentially bounded functions space. Firstly, stabilization of solutions to nonlinear impulsive evolution equations are studied by means of fixed point methods at an appropriate decay rate. Secondly, stable manifolds for the associated singular perturbation problems with impulses are compared with each other. Finally, an example on initial boundary value problem for impulsive parabolic equations is illustrated to our theory results.},
author = {Wang, JinRong, Zhou, Yong, Wei, Wei},
journal = {Kybernetika},
keywords = {impulsive evolution equations; stabilization; stable manifolds; singularly perturbed problems; impulsive evolution equations; stable manifolds; singularly perturbed problems},
language = {eng},
number = {6},
pages = {1211-1228},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Novel method for generalized stability analysis of nonlinear impulsive evolution equations},
url = {http://eudml.org/doc/251392},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Wang, JinRong
AU - Zhou, Yong
AU - Wei, Wei
TI - Novel method for generalized stability analysis of nonlinear impulsive evolution equations
JO - Kybernetika
PY - 2012
PB - Institute of Information Theory and Automation AS CR
VL - 48
IS - 6
SP - 1211
EP - 1228
AB - In this paper, we discuss some generalized stability of solutions to a class of nonlinear impulsive evolution equations in the certain piecewise essentially bounded functions space. Firstly, stabilization of solutions to nonlinear impulsive evolution equations are studied by means of fixed point methods at an appropriate decay rate. Secondly, stable manifolds for the associated singular perturbation problems with impulses are compared with each other. Finally, an example on initial boundary value problem for impulsive parabolic equations is illustrated to our theory results.
LA - eng
KW - impulsive evolution equations; stabilization; stable manifolds; singularly perturbed problems; impulsive evolution equations; stable manifolds; singularly perturbed problems
UR - http://eudml.org/doc/251392
ER -

References

top
  1. Abada, N., Benchohra, M., Hammouche, H., 10.1016/j.jde.2009.03.004, J. Differential Equations 246 (2009), 3834-3863. Zbl1171.34052MR2514728DOI10.1016/j.jde.2009.03.004
  2. Ahmed, N. U., 10.1137/S0363012901391299, SIAM J. Control Optim. 42 (2003), 669-685. MR1982287DOI10.1137/S0363012901391299
  3. Ahmed, N. U., Teo, K. L., Hou, S. H., 10.1016/S0362-546X(03)00117-2, Nonlinear Anal. 54 (2003), 907-925. Zbl1030.34056MR1992511DOI10.1016/S0362-546X(03)00117-2
  4. Benchohra, M., Henderson, J., Ntouyas, S. K., Impulsive differential equations and inclusions., In: Contemporary Mathematics and Its Applications, Vol. 2, Hindawi Publishing Corporation, New York 2006. Zbl1130.34003MR2322133
  5. Bounit, H., Hammouri, H., 10.1007/s002459900075, Appl. Math. Optim. 37 (1998), 225-242. Zbl1041.93553MR1489316DOI10.1007/s002459900075
  6. Chang, Y. K., Nieto, J. J., 10.1080/01630560902841146, Numer. Funct. Anal. Optim. 30 (2009), 227-244. Zbl1176.34096MR2514215DOI10.1080/01630560902841146
  7. Dvirnyĭ, A. I., Slyn'ko, V. I., Stability of solutions to impulsive differential equations in critical cases., Sibirsk. Mat. Zh. 52 (2011), 70-80. MR2810251
  8. Fan, Z., Li, G., 10.1016/j.jfa.2009.10.023, J. Funct. Anal. 258 (2010), 1709-1727. Zbl1193.35099MR2566317DOI10.1016/j.jfa.2009.10.023
  9. Hernández, E., Rabello, M., Henríquez, H. R., 10.1016/j.jmaa.2006.09.043, J. Math. Anal. Appl. 331 (2007), 1135-1158. MR2313705DOI10.1016/j.jmaa.2006.09.043
  10. Koliha, J. J., Straškraba, I., Stability in nonlinear evolution problems by means of fixed point theorem., Comment. Math. Univ. Carolin. 38 (1997), 37-59. MR1455469
  11. Lakshmikantham, V., Bainov, D. D., Simeonov, P. S., Theory of Impulsive Differential Equations., World Scientific, Singapore - London 1989. Zbl0719.34002MR1082551
  12. Liang, J., Liu, J. H., Xiao, T.-J., 10.1016/j.mcm.2008.05.046, Math. Comput. Model. 49 (2009), 798-804. Zbl1173.34048MR2483682DOI10.1016/j.mcm.2008.05.046
  13. Liu, J., Nonlinear impulsive evolution equations., Dynamic Contin. Discrete Impuls. Syst. 6 (1999), 77-85. Zbl0932.34067MR1679758
  14. Lü, J., Chen, G., 10.1142/S0218127406015179, Internat. J. Bifurcation and Chaos 16 (2006), 775-858. MR2234259DOI10.1142/S0218127406015179
  15. Lü, J., Han, F., Yu, X., Chen, G., 10.1016/j.automatica.2004.06.001, Automatica 40 (2004), 1677-1687. Zbl1162.93353MR2155461DOI10.1016/j.automatica.2004.06.001
  16. Wang, J., Dong, X., Zhou, Y., 10.1016/j.cnsns.2011.05.034, Comm. Nonlinear Sci. Numer. Simul. 17 (2012), 545-554. MR2834413DOI10.1016/j.cnsns.2011.05.034
  17. Wang, J., Dong, X., Zhou, Y., 10.1016/j.cnsns.2011.12.002, Comm. Nonlinear Sci. Numer. Simul. 17 (2012), 3129-3139. MR2834413DOI10.1016/j.cnsns.2011.12.002
  18. Wang, J., Wei, W., 10.1007/s00025-010-0057-x, Results Math. 58 (2010), 379-397. Zbl1209.34095MR2728164DOI10.1007/s00025-010-0057-x
  19. Wei, W., Xiang, X., Peng, Y., 10.1080/02331930500530401, Optimization 55 (2006), 141-156. MR2221729DOI10.1080/02331930500530401
  20. Wang, J., Xiang, X., Peng, Y., 10.1016/j.na.2009.01.139, Nonlinear Anal. 71 (2009), e1344-e1353. Zbl1238.34079MR2671921DOI10.1016/j.na.2009.01.139
  21. Wei, W., Hou, S., Teo, K. L., On a class of strongly nonlinear impulsive differential equation with time delay., Nonlinear Dyn. Syst. Theory 6 (2006), 281-293. Zbl1114.47067MR2264177
  22. Xiang, X., Wei, W., Jiang, Y., Strongly nonlinear impulsive system and necessary conditions of optimality., Dyn. Cont. Discrete Impuls. Syst. 12 (2005), 811-824. Zbl1081.49027MR2178682
  23. Xu, D., Yang, Z., Yang, Z., 10.1016/j.na.2006.07.043, Nonlinear Anal. 67 (2007), 1426-1439. Zbl1122.34063MR2323290DOI10.1016/j.na.2006.07.043
  24. Yang, T., Impulsive Control Theory., Springer-Verlag, Berlin - Heidelberg 2001. Zbl0996.93003MR1850661
  25. Yang, Z., Xu, D., 10.1109/TAC.2007.902748, IEEE Trans. Automat. Control 52 (2007), 1148-1154. MR2342720DOI10.1109/TAC.2007.902748
  26. Yu, X., Xiang, X., Wei, W., 10.1016/j.jmaa.2006.03.075, J. Math. Anal. Appl. 327 (2007), 220-232. MR2277406DOI10.1016/j.jmaa.2006.03.075
  27. Zhang, Y., Sun, J., 10.1016/j.jmaa.2004.07.018, J. Math. Anal. Appl. 301 (2005), 237-248. Zbl1068.34073MR2105932DOI10.1016/j.jmaa.2004.07.018

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.