Abstract

We study stability and attraction for a nonlinear dynamical system with nonsmooth Lyapunov functions. The previous results on stability and attraction with a max-type Lyapunov function are extended to the case where Lyapunov function is piecewise smooth. A condition, under which stability and attraction are guaranteed with a piecewise smooth Lyapunov function, is proposed. Taking two certain classes of piecewise smooth functions as Lyapunov functions, related conditions for stability and attraction are developed.

Author information

Contact details are reproduced from the original publication and may be historical.

Yan Gao

School of Management, University of Shanghai for Science and Technology, Shanghai 200093, P. R. China

gaoyan@usst.edu.cn

Keywords

  • Nonlinear dynamical system
  • stability
  • region of attraction
  • Lyapnov functions
  • nonsmooth analysis
  • piecewise smooth function

Mathematics Subject Classification

90C30, 65H10

References

  1. J.-P. Aubin: Optima and Equilibria: An Introduction to Nonlinear Analysis, Springer, Berlin (1993).https://doi.org/10.1007/978-3-662-02959-6Google Scholar
  2. R. W. Chaney: Piecewise C k functions in nonsmooth analysis, Nonlinear Anal. 15 (1990) 649–660.https://doi.org/10.1016/0362-546X(90)90005-2Google Scholar
  3. F. H. Clarke, Y. S. Ledyaev, R. J. Stern, F. R. Wolenski: Nonsmooth Analysis and Control Theory, Springer, New York (1998).Google Scholar
  4. Y. Gao, J. Lygeros, M. Quincampoix, N. Seube: On the control of uncertain impulsive systems: approximate stabilisation and controlled invariance, Int. J. Control 77 (2004) 1393–1407.https://doi.org/10.1080/00207170412331317431Google Scholar
  5. A. Hassibi, S. Boyd: Quadratic stabilization and control of piecewise linear system, Proc. American Control Conference (1998) 3659–3664.https://doi.org/10.1109/ACC.1998.703296Google Scholar
  6. T. Hu, Z. Lin: Composite quadratic Lyapunov functions for constrained control systems, IEEE Trans. Autom. Control 48 (2003) 440–450.https://doi.org/10.1109/TAC.2003.809149Google Scholar
  7. T. Hu, L. Ma, Z. Lin: Stabilization of switched system via composite quadratics functions, IEEE Trans. Autom. Control 53 (2008) 2571–2585.https://doi.org/10.1109/TAC.2008.2006933Google Scholar
  8. J. S. Pang, D. Ralph: Piecewise smoothness, local invertibility and parametric analysis of normal maps, Math. Oper. Res. 21 (1996) 401–426.https://doi.org/10.1287/moor.21.2.401Google Scholar
  9. M. Quincampoix, N. Seube: Stabilization of uncertain control systems through piecewise constant feedback, J. Math. Anal. Appl. 218 (1998) 240–255.https://doi.org/10.1006/jmaa.1997.5775Google Scholar
  10. W. Tan, A. Packard: Stability region analysis using polynomial and composite polynomial Lyapunov functions and sum-of squares programming, IEEE Trans. Autom. Control 53(2) (2008) 565–571.https://doi.org/10.1109/TAC.2007.914221Google Scholar
  11. L. Xie, S. Shishkin, M. Fu: Piecewise Lyapunov functions for robust stability of linear time-varying systems, Syst. Control Lett. 31 (1997) 165–171.https://doi.org/10.1016/S0167-6911(97)00027-3Google Scholar

Suggested citation

Y. Gao. “Piecewise Smooth Lyapunov Function for a Nonlinear Dynamical System.” Journal of Convex Analysis 19 (2012), No. 4, 1009–1015. https://doi.org/10.68381/jca19053

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Published by Heldermann Verlag, 2012. Rights now held by Banach Press.