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http://dx.doi.org/10.4134/BKMS.2014.51.1.001

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)
Publication Information
Bulletin of the Korean Mathematical Society / v.51, no.1, 2014 , pp. 1-11 More about this Journal
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.
Keywords
switched systems; control systems; asymptotically controllable; control-Lyapunov function; differential inclusions;
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1 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.
2 A. A. Agrachev and D. Liberzon, Lie-algebraic stability criteria for switched systems, SIAM J. Control Optim. 40 (2001), no. 1, 253-269.   DOI   ScienceOn
3 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.
4 J. P. Aubin, Viability Theory, Birkhauser Boston, Inc., Boston, MA, 1991.
5 J. P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin, 1984.
6 J. P. Aubin and H. Frankowska, Set-Valued Analysis, Springer-Verlag, Birkhauser Boston Basel Berlin, 1990.
7 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.   DOI   ScienceOn
8 R. Gilmore, Lie Groups, Lie Algebras and Some of Their Applications, John Wiley, New York, 1994.
9 Y. S. Ledyaev and E. D. Sontag, A Lyapunov characterization of robust stabilization, Nonlinear Anal. 37 (1999), no. 7, 813-840.   DOI   ScienceOn
10 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.
11 P. J. Olver, Applications of Lie Groups to Differential Equations, 2nd ed. New York, Springer-Verlag, 1993.
12 E. D. Sontag, A Lyapunov-like characterization of asymptotic controllability, SIAM J. Control Optim. 21 (1983), no. 3, 462-471.   DOI   ScienceOn