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Strict Lyapunov functions for impulsive hybrid time-varying systems with discontinuous right-hand side

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Abstract

In this paper, explicit closed form expressions of nonsmooth strict Lyapunov functions for impulsive hybrid time-varying systems with discontinuous right-hand side is provided. Lyapunov functions are expressed in terms of known nonstrict Lyapunov functions for the dynamics and finite sums of persistency of excitation parameters.

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References

  1. A. M. Samoilenko and N. A. Perestyuk, Differential Equations with Impusive Effect, Visca Skola, 1987.

  2. G. K. Kulev and D. D. Bainov, Strong stability of impulsive systems, Int. J. Theor. Phys., 1988, 27: 745–755.

    Article  MATH  MathSciNet  Google Scholar 

  3. D. D. Bainov, A. D. Myshkis, and G. T. Stamov, Dichotomies and almost periodicity of the solutions of systems of impulsive differential equations, Dyn. Syst. Appl., 1996, 5: 145–152.

    MATH  MathSciNet  Google Scholar 

  4. D. D. Bainov, A. D. Myshkis, and G. T. Stamov, Almost periodic solutions of hyperbolic systems of impulsive differential equation, Kumamoto J. Math, 1997, 10: 1–10.

    MATH  MathSciNet  Google Scholar 

  5. Z. G. Li, C. Y. Wen, and Y. C. Soh, A unified approach for stability analysis of impusive hybrid systems, Proceedings of the 38th Conference on Decision Control, Phoenix, Arizons, USA, 1999.

  6. G.T. Stamov, Existence of almost periodic solutions for strong stable impulsive differential equations, IMA Journal of Mathematical Control and Information, 2001, 18: 153–160.

    Article  MATH  MathSciNet  Google Scholar 

  7. G. N. Silva and F. L. Pereira, Lyapunov stability for impulsive dynamical systems, Proceedings of the 41st IEEE Conference on decision and control, Las Vegas, Nevada USA, December, 2002.

  8. N. Zhang and T. J. Wu, Guaranteed performance control for uncertain impulsive hybrid systems and its application, Proceedings of the American Control Conference, Anchorage, AK May, 2002: 8–10.

  9. Y. J. Zhang, B. Liu, and L. S. Chen, Extinction and permanence of a two-prey one-predator system with impulsive effect, Mathematical Medicine and Biology, 2003, 20: 309–325.

    Article  MATH  Google Scholar 

  10. B. Liu and L. S. Chen, The periodic competing Lotka-Volterra model with impulsive effct, Mathematical Medicine and Biology, 2004, 21: 129–145.

    Article  MATH  Google Scholar 

  11. J. P. Hespanha, D. Liberzon, and A. R. Teel, On input-to-state stability of impulsive systems, Proceedings of 44th IEEE Conference on Decesion and Control, and the European Control Conference, Seville, Spain, December, 2005: 12–15.

  12. J. Yao, Z. H. Guan, G. R. Chen, et al., Stability, robust stabilization and H ∞ control of singular-impulsive systems via switching control, Systems and Control Letters, 2006, 55: 879–886.

    Article  MATH  MathSciNet  Google Scholar 

  13. T. Donchev, Impulsive differential inclusion with constrains, Electronic Journal of Differential Equations, 2006, 66: 1–12.

    MathSciNet  Google Scholar 

  14. A. Bressan, Impulsive control of Lagrangian systems and locomotion in fluids, Discrete and Continuous Dynamical Systems, 2008, 20(1): 1–35.

    Article  MATH  MathSciNet  Google Scholar 

  15. S. A. Belbas and W. H. Schmidt, Optimal control of Volterra equations with impulses, Applied Mathematics and Computation, 2005, 166: 696–723.

    Article  MATH  MathSciNet  Google Scholar 

  16. F. Albertini and E. D. Sontag, Continuous control-Lyapunov functions for asymptotically controllable time-varying systems, International Journal of Control, 1999, 7: 1630–1641.

    Article  MathSciNet  Google Scholar 

  17. D. Angeli and E. D. Sontag, Forward completeness, unboundedness observability, and their Lyapunov characterizations, Systems and Control Letters, 1999, 38: 209–217.

    Article  MATH  MathSciNet  Google Scholar 

  18. F. Mazenc, Strict Lyapunov functions for time-varying systems, Automatica, 2003, 39: 349–353.

    Article  MATH  MathSciNet  Google Scholar 

  19. F. Mazenc and Nesic, Lyapunov functions for time varying systems satisfying generalized conditions of Matrosov theorem, Proceedings of the 44th IEEE Conference on Decision and Control (CDC) and European Control Conference ECC 05, Seville, Spain, December, 2005: 2432–2437.

  20. M. Malisoff and F. Mazenc, Constructions of strict Lyapunov functions for discrete time and hybrid time-varying systems, Nonlinear Analysis: Hybrid Systems and Applications, 2006.

  21. M. Malisoff, L. Rifford, and E. D. Sontag, Global asymptotic controllability and input-to-state stabilization: The effect of actuator errors, Optimal Control, Stabilization, and Nonsmooth Analysis, Lecture Notes in Control and Inform. Sci., Springer-Verlag, Heidelberg, 2004, 301: 155–171.

    Google Scholar 

  22. E. D. Sontag, The ISS philosophy as a unifying framework for stability-like behavior, Nonlinear Control in the Year 2000, Vol. 2, Lecture Notes in Control and Inform. Sci., Springer, London. 2001, 259: 443–467.

    Google Scholar 

  23. A. F. Filippov, Differential equations with discontinuous right-hand side, Amer. Math. Soc. Translations, 1998, 42(2): 191–231.

    Google Scholar 

  24. A. F. Filippov, Differential Equations with Discontinuous Right Hand Sides, Kluwer, Boston, 1988.

    MATH  Google Scholar 

  25. D. Shevitz and B. Paden, Lyapunov stability theory of nonsmooth systems, IEEE Transactions on Automatic Control, 1994, 39(9).

  26. A. Bacciotti and F. Ceragioli, L-2-gain stabilizability with respect to Filippov solutions, Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, Nevada USA, December, 2002.

  27. M. Malisoff and F. Mazenc, Further remarks on strict input-to-state stable Lyapunov functions for time-varying systems, Automatica, 2005, 41: 1973–1978.

    Article  MATH  MathSciNet  Google Scholar 

  28. F. Mazenc, and M. Malisoff, Further constructions of control Lyapunov functions and stabilizing feedbacks for systems satisfying the Jurdjevic-Quinn conditions, IEEE Transactions on Automatic Control, 2006, 51: 360–365.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Zhengzhou University, Zhengzhou, Henan, 450001, China

    Xiaowu Mu & Fengjun Tang

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  1. Xiaowu Mu
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  2. Fengjun Tang
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Correspondence to Xiaowu Mu.

Additional information

This research is supported by the National Natural Science Foundation of China under Grant No. 60874006.

This paper was recommended for publication by Editor Jifeng ZHANG.

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Mu, X., Tang, F. Strict Lyapunov functions for impulsive hybrid time-varying systems with discontinuous right-hand side. J Syst Sci Complex 24, 261–270 (2011). https://doi.org/10.1007/s11424-011-7237-y

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  • DOI: https://doi.org/10.1007/s11424-011-7237-y

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