# A LYAPUNOV CHARACTERIZATION OF ASYMPTOTIC CONTROLLABILITY FOR NONLINEAR SWITCHED SYSTEMS

• Wang, Yanling (Department of Mathematics Tianjin University) ;
• Qi, Ailing (College of Science Civil Aviation University of China)
• Published : 2014.01.31

#### Abstract

In this paper, we show that general nonlinear switched systems are asymptotically controllable if and only if there exist control-Lyapunov functions for their relaxation systems. If the switching signal is dependent on the time, then the control-Lyapunov functions are continuous. And if the switching signal is dependent on the state, then the control-Lyapunov functions are $C^1$-smooth. We obtain the results from the viewpoint of control system theory. Our approach is based on the relaxation theorems of differential inclusions and the classic Lyapunov characterization.

#### References

1. A. A. Agrachev and D. Liberzon, Lie-algebraic stability criteria for switched systems, SIAM J. Control Optim. 40 (2001), no. 1, 253-269. https://doi.org/10.1137/S0363012999365704
2. P. J. Antsaklis, J. A. Stiver, M. D. Lemmon et al., Hybrid Systems, Vol. 736 of Lecture Notes in Computer Science, 366-392, Heidelberg, Springer, 1993.
3. J. P. Aubin, Viability Theory, Birkhauser Boston, Inc., Boston, MA, 1991.
4. J. P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin, 1984.
5. J. P. Aubin and H. Frankowska, Set-Valued Analysis, Springer-Verlag, Birkhauser Boston Basel Berlin, 1990.
6. F. H. Clarke, Y. S. Ledyaev, and R. J. Stern, Asymptotic stability and smooth Lyapunov functions, J. Differential Equations 149 (1998), no. 1, 69-114. https://doi.org/10.1006/jdeq.1998.3476
7. R. Gilmore, Lie Groups, Lie Algebras and Some of Their Applications, John Wiley, New York, 1994.
8. Y. S. Ledyaev and E. D. Sontag, A Lyapunov characterization of robust stabilization, Nonlinear Anal. 37 (1999), no. 7, 813-840. https://doi.org/10.1016/S0362-546X(98)00075-3
9. A. M. Lyapunov, The general problem of the stability of motion, Math. Soc. Kharkov,1892 (Russian); English Translation: Internat. J. Control 55 (1992), 531-773.
10. P. J. Olver, Applications of Lie Groups to Differential Equations, 2nd ed. New York, Springer-Verlag, 1993.
11. E. D. Sontag, A Lyapunov-like characterization of asymptotic controllability, SIAM J. Control Optim. 21 (1983), no. 3, 462-471. https://doi.org/10.1137/0321028
12. E. D. Sontag and H. J. Sussmann, Nonsmooth control Lyapunov functions, Proc. IEEE Conf. Decision and Control, New Orleans, Dec. 1995, IEEE Publications, 2799-2805, 1995.