Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                

Kybernetika 48 no. 6, 1211-1228, 2012

Novel method for generalized stability analysis of nonlinear impulsive evolution equations

JinRong Wang, Yong Zhou and Wei Wei

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.

Keywords:

stabilization, impulsive evolution equations, stable manifolds, singularly perturbed problems

Classification:

34G20, 35B40, 35K20

paper.pdf

References:

  1. N. Abada, M. Benchohra and H. Hammouche: Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions. J. Differential Equations 246 (2009), 3834-3863.   CrossRef
  2. N. U. Ahmed: Existence of optimal controls for a general class of impulsive systems on Banach space. SIAM J. Control Optim. 42 (2003), 669-685.   CrossRef
  3. N. U. Ahmed, K. L. Teo and S. H. Hou: Nonlinear impulsive systems on infinite dimensional spaces. Nonlinear Anal. 54 (2003), 907-925.   CrossRef
  4. M. Benchohra, J. Henderson and S. K. Ntouyas: Impulsive differential equations and inclusions. In: Contemporary Mathematics and Its Applications, Vol. 2, Hindawi Publishing Corporation, New York 2006.   CrossRef
  5. H. Bounit and H. Hammouri: Stabilization of infinite-dimensional semilinear systems with dissipative drift. Appl. Math. Optim. 37 (1998), 225-242.   CrossRef
  6. Y. K. Chang and J. J. Nieto: Existence of solutions for impulsive neutral integro-differential inclusions with nonlocal initial conditions via fractional operators. Numer. Funct. Anal. Optim. 30 (2009), 227-244.   CrossRef
  7. A. I. Dvirny\u{i} and V. I. Slyn'ko: Stability of solutions to impulsive differential equations in critical cases. Sibirsk. Mat. Zh. 52 (2011), 70-80.   CrossRef
  8. Z. Fan and G. Li: Existence results for semilinear differential equations with nonlocal and impulsive conditions. J. Funct. Anal. 258 (2010), 1709-1727.   CrossRef
  9. E. Hernández, M. Rabello and H. R. Henríquez: Existence of solutions for impulsive partial neutral functional differential equations. J. Math. Anal. Appl. 331 (2007), 1135-1158.   CrossRef
  10. J. J. Koliha and I. Stra\u{s}kraba: Stability in nonlinear evolution problems by means of fixed point theorem. Comment. Math. Univ. Carolin. 38 (1997), 37-59.   CrossRef
  11. V. Lakshmikantham, D. D. Bainov and P. S. Simeonov: Theory of Impulsive Differential Equations. World Scientific, Singapore - London 1989.   CrossRef
  12. J. Liang, J. H. Liu and T.-J. Xiao: Nonlocal impulsive problems for nonlinear differential equations in Banach spaces. Math. Comput. Model. 49 (2009), 798-804.   CrossRef
  13. J. Liu: Nonlinear impulsive evolution equations. Dynamic Contin. Discrete Impuls. Syst. 6 (1999), 77-85.   CrossRef
  14. J. Lü and G. Chen: Generating multiscroll Chaotic attractors: Theories, Methods and Applications. Internat. J. Bifurcation and Chaos 16 (2006), 775-858.   CrossRef
  15. J. Lü, F. Han, X. Yu and G. Chen: Generating 3-D multi-scroll chaotic attractors: A hysteresis series switching method. Automatica 40 (2004), 1677-1687.   CrossRef
  16. J. Wang, X. Dong and Y. Zhou: Existence, attractiveness and stability of solutions for quadratic Urysohn fractional integral equations. Comm. Nonlinear Sci. Numer. Simul. 17 (2012), 545-554.   CrossRef
  17. J. Wang, X. Dong and Y. Zhou: Analysis of nonlinear integral equations with Erdélyi-Kober fractional operator. Comm. Nonlinear Sci. Numer. Simul. 17 (2012), 3129-3139.   CrossRef
  18. J. Wang and W. Wei: A class of nonlocal impulsive problems for integrodifferential equations in Banach spaces. Results Math. 58 (2010), 379-397.   CrossRef
  19. W. Wei, X. Xiang and Y. Peng: Nonlinear impulsive integro-differential equation of mixed type and optimal controls. Optimization 55 (2006), 141-156.   CrossRef
  20. J. Wang, X. Xiang and Y. Peng: Periodic solutions of semilinear impulsive periodic system on Banach space. Nonlinear Anal. 71 (2009), e1344-e1353.   CrossRef
  21. W. Wei, S. Hou and K. L. Teo: On a class of strongly nonlinear impulsive differential equation with time delay. Nonlinear Dyn. Syst. Theory 6 (2006), 281-293.   CrossRef
  22. X. Xiang, W. Wei and Y. Jiang: Strongly nonlinear impulsive system and necessary conditions of optimality. Dyn. Cont. Discrete Impuls. Syst. 12 (2005), 811-824.   CrossRef
  23. D. Xu, Z. Yang and Z. Yang: Exponential stability of nonlinear impulsive neutral differential equations with delays. Nonlinear Anal. 67 (2007), 1426-1439.   CrossRef
  24. T. Yang: Impulsive Control Theory. Springer-Verlag, Berlin - Heidelberg 2001.   CrossRef
  25. Z. Yang and D. Xu: Stability analysis and design of impulsive control system with time delay. IEEE Trans. Automat. Control 52 (2007), 1148-1154.   CrossRef
  26. X. Yu, X. Xiang and W. Wei: Solution bundle for class of impulsive differential inclusions on Banach spaces. J. Math. Anal. Appl. 327 (2007), 220-232.   CrossRef
  27. Y. Zhang and J. Sun: Strict stability of impulsive functional differential equations. J. Math. Anal. Appl. 301 (2005), 237-248.   CrossRef