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converse Lyapunov theorem; practical asymptotic stability; impulsive systems; cascade systems; perturbed systems
The Lyapunov's second method is one of the most famous techniques for studying the stability properties of dynamic systems. This technique uses an auxiliary function, called Lyapunov function, which checks the stability properties of a specific system without the need to generate system solutions. An important question is about the reversibility or converse of Lyapunov's second method; i. e., given a specific stability property does there exist an appropriate Lyapunov function? The main result of this paper is a converse Lyapunov Theorem for practical asymptotic stable impulsive systems. Applying our converse Theorem, several criteria on practical asymptotic stability of the solution of perturbed impulsive systems and cascade impulsive systems are established. Finally, some examples are given to show the effectiveness of the derived results.
[1] Bacciotti, A., Rosier, L.: Lyapunov and Lagrange stability: Inverse theorems for discontinuous systems. Math. Control Signals Systems 11 (1998), 101-128. DOI 10.1007/bf02741887 | MR 1628047
[2] Bainov, D. D., Simeonov, P. S.: Systems with Impulse Effect: Stability, Theory, and Applications. Ellis Horwood, Chichester 1989. MR 1010418
[3] Benabdallah, A., Ellouze, I., Hammami, M. A.: Practical stability of nonlinear time-varying cascade systems. J. Dynamical Control Systems 15 (2009), 45-62. DOI 10.1007/s10883-008-9057-5 | MR 2475660
[4] Benabdallah, A., Dlala, M., Hammami, M. A.: A new Lyapunov function for stability of time-varying nonlinear perturbed systems. Systems Control Lett. 56 (2007), 179-187. DOI 10.1016/j.sysconle.2006.08.009 | MR 2296644
[5] Hamed, B. Ben, Ellouze, I., Hammami, M. A.: Practical uniform stability of nonlinear differential delay equations. Mediterranean J. Math. 8 (2011), 603-616. DOI 10.1007/s00009-010-0083-7 | MR 2860688
[6] Hamed, B. Ben, Hammami, M. .A: Practical stabilization of a class of uncertain time-varying nonlinear delay systems. J. Control Theory Appl. 7 (2009), 175-180. DOI 10.1007/s11768-009-8017-2 | MR 2526947
[7] Cai, C., Teel, A., Goebel, R.: Smooth Lyapunov functions for hybrid systems, Part I: Existence is equivalent to robustness. IEEE Trans. Automat. Control 52 (2007), 7, 1264-1277. DOI 10.1109/tac.2007.900829 | MR 2332751
[8] Corless, M.: Guaranteed rates of exponential convergence for uncertain systems. J. Optim. Theory Appl. 64 (1990), 481-494. DOI 10.1007/bf00939420 | MR 1043736
[9] Dlala, M., Ghanmi, B., Hammami, M. A: Exponential practical stability of nonlinear impulsive systems: converse theorem and applications. Dynamics Continuous Discrete Impulsive Systems 21 (2014), 37-64. MR 3202437
[10] Dlala, M., Hammami, M. A.: Uniform exponential practical stability of impulsive perturbed systems. J. Dynamical Control Systems 13 (2007), 373-386. DOI 10.1007/s10883-007-9020-x | MR 2337283
[11] Giesl, P., Hafstein, S.: Review on computational methods for Lyapunov functions. Discrete Continuous Dynamical Systems: Series B 20 (2015), 2291-2331. DOI 10.3934/dcdsb.2015.20.2291 | MR 3423237
[12] Gordon, S. P.: On converse to the stability theorems for difference equations. SIAM J. Control Optim. 10 (1972), 76-81. DOI 10.1137/0310007 | MR 0318707
[13] Hahn, W.: Stability of Motion. Springer-Verlag, 1967. DOI 10.1007/978-3-642-50085-5 | MR 0223668 | Zbl 0189.38503
[14] Isidori, A.: Nonlinear Control Systems. Second edition. Springer-Verlag, 1989. DOI 10.1007/978-3-662-02581-9 | MR 1015932
[15] Kellett, C.: Converse Theorems in lyapunov's second method. Discrete Continuous Dynamical Systems: Series B 20 (2015), 2333-2360. DOI 10.3934/dcdsb.2015.20.2333 | MR 3423238
[16] Jiang, Z. P., Teel, A. R., Praly, L.: Small gain theorem for ISS systems and applications. Math. Control, Signals Systems 7 (1995), 95-120. DOI 10.1007/bf01211469 | MR 1359023
[17] Wang, Y.: A converse Lyapunov theorem for discrete-time systems with disturbances. Systems Control Lett, 45 (2002), 49-58. DOI 10.1016/s0167-6911(01)00164-5 | MR 2010491
[18] Khalil, H. K.: Nonlinear Systems. Third edition. Macmillan Publishing Company, 2002. MR 1201326
[19] Lakshmikantham, V., Leela, S., Martynyuk, A. A.: Practical Stability of Nonlinear Systems. World Scientific, Singapore 1990. DOI 10.1142/1192 | MR 1089428
[20] Lakshmikantham, V., Bainov, D. D., Simeonov, P. S.: Theory of Impulsive Differential Equations. Series in Modern Applied Mathematics. Singapore and Teaneck, World Scientific, NJ 1989. DOI 10.1142/0906 | MR 1082551
[21] LaSalle, J. P., Lefschetz, S.: Stability by Lyapunov's Direct Method with Applications. Academic Press, New York 1961. DOI 10.1002/zamm.19620421022 | MR 0132876
[22] Lin, Y., Sontag, E. D., Wang, Y.: A smooth converse Lyapunov theorem for robust stability. SIAM J. Control Optim. 34 (1996), 124-160. DOI 10.1137/s0363012993259981 | MR 1372908
[23] Mancilla-Aguilar, J. L., Garcia, R. A.: A converse Lyapunov theorem for nonlinear switch systems. Systems Control Lett. 41 (2000), 67-71. DOI 10.1016/s0167-6911(00)00040-2 | MR 1827722
[24] Panteley, E., Loria, A.: On global uniform asymptotic stability of non linear time-varying non autonomous systems in cascade. Systems Control Lett. 33 (1998), 131-138. DOI 10.1016/s0167-6911(97)00119-9 | MR 1607815
[25] Pradalier, C., Siegwart, R., Hirzinger, G.: Robotics Research. Springer-Verlag, Berlin 2011. DOI 10.1007/978-3-642-19457-3
[26] Spong, M. W., Vidyasagar, M.: Robot Dynamics and Control. John Wiley and Sons, Inc, New York 1989.
[27] Spong, M. W.: The control of underactuated mechanical systems. In: First International Conference on Mecatronics, Mexico City 1994.
[28] Tsinias, J.: A converse Lyapunov theorem for nonuniform in time, global exponential robust stability. Systems Control Lett. 44 (2001), 373-384. DOI 10.1007/978-3-642-19457-3 | MR 2021956
[29] Yang, X.-S.: Existence of unbounded solutions of time varying systems and failure of global asymptotic stability in discrete-time cascade systems. IMA J. Math. Control Inform. 22 (2005), 80-87. DOI 10.1093/imamci/dni006 | MR 2122276
[30] Yang, T.: Impulsive Control Theory. Springer, 2001. DOI 10.1007/3-540-47710-1 | MR 1850661 | Zbl 0996.93003
[31] Yoshizawa, T.: Stability Theory by Lyapunov's Second Method. Mathematical Society of Japan, 1966. MR 0208086
[32] Zubov, V. I.: Methods of A. M. Lyapunov and their Application. P. Noordhoff Ltd, Groningen 1964; translated from the Russian edition of 1957. MR 0179428
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