# American Institute of Mathematical Sciences

December  2011, 31(4): 1023-1038. doi: 10.3934/dcds.2011.31.1023

## Finite-time Lyapunov stability analysis of evolution variational inequalities

 1 Université de La Réunion, PIMENT EA 4518, 97400 Saint-Denis, France 2 XLIM UMR-CNRS 6172, Université de Limoges, 87060 Limoges, France, France

Received  November 2009 Revised  October 2010 Published  September 2011

Using Lyapunov's stability and LaSalle's invariance principle for nonsmooth dynamical systems, we establish some conditions for finite-time stability of evolution variational inequalities. The theoretical results are illustrated by some examples drawn from electrical circuits involving nonsmooth elements like diodes.
Citation: Khalid Addi, Samir Adly, Hassan Saoud. Finite-time Lyapunov stability analysis of evolution variational inequalities. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1023-1038. doi: 10.3934/dcds.2011.31.1023
##### References:
 [1] K. Addi, S. Adly, B. Brogliato and D. Goeleven, A method using the approach of Moreau and Panagiotopoulos for the mathematical formulation of non-regular circuits in electronics, Nonlinear Analysis C: Hybrid Systems and Applications, 1 (2007), 30-43. [2] S. Adly, Attractivity theory for second order non-smooth dynamical systems with application to dry friction, Journal of Mathematical Analysis and Applications, 322 (2006), 1055-1070. doi: 10.1016/j.jmaa.2005.09.076. [3] S. Adly and D. Goeleven, A stability theory for second-order nonsmooth dynamical systems with application to friction problems, J. Maths. Pures Appl., 83 (2004), 17-51. doi: 10.1016/S0021-7824(03)00071-0. [4] J.-P. Aubin and A. Cellina, "Differential Inclusions. Set-Valued Maps and Viability Theory," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1984. [5] J. Alvarez, I. Orlov and L. Acho, An invariance principle for discontinuous dynamic systems with applications to a Coulomb friction oscillator, Journal of Dynamic Systems, Measurement, and Control, 122 (2000), 687-690. doi: 10.1115/1.1317229. [6] S. P. Bhat and D. S. Bernstein, Finite-time stability of continuous autonomous systems, SIAM J. Control Optimization, 38 (2000), 751-766. doi: 10.1137/S0363012997321358. [7] F. H. Clarke, "Optimization and Nonsmooth Analysis," Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, New York, 1983. [8] A. F. Filippov, "Differential Equations with Discontinuous Right-Hand Sides," Kluwer, Dordrecht, The Netherlands, 1988. [9] D. Goeleven, D. Motreanu and V. V. Motreanu, On the stability of stationary solutions of first order evolution variational inequalities, Adv. Nonlinear Var. Inequal., 6 (2003), 1-30. [10] D. Goeleven and B. Brogliato, Necessary conditions of asymptotic stability for unilateral dynamical systems, Nonlinear Anal., 61 (2005), 961-1004. doi: 10.1016/j.na.2005.01.037. [11] D. Goeleven and B. Brogliato, Stability and unstability matrices for linear evolution variational inequalities, submitted, 2002. [12] K. K. Hassan, "Nonlinear Systems," Prentice-Hall, Upper Saddle River, NJ, 1996. [13] V. T. Haimo, Finite time controllers, SIAM J. Control and Optimisation, 24 (1986), 760-770. [14] T. Kato, Accretive operators and nonlinear evolutions equations in Banach spaces, in "Nonlinear Functional Analysis" (Proc. Sympos. Pure Math., Vol. XVIII, Part 1, Chicago, Ill., 1968), Amer. Math. Society, Providence, RI, 1970. [15] M. Kocan and P. Soravia, Lyapunov functions for infnite-dimensional systems, J. Funct. Anal., 192 (2002), 342-–363. doi: 10.1006/jfan.2001.3910. [16] E. Moulay and W. Perruquatti, Finite time stability of differential inclusions, IMA J. Math. Control Info., 22 (2005), 465-475. doi: 10.1093/imamci/dni039. [17] E. Moulay and W. Perruquatti, Finite time stability and stabilization of a class of continuous systems, J. Math. Anal. Appli., 323 (2006), 1430-1443. doi: 10.1016/j.jmaa.2005.11.046. [18] Y. Orlov, Finite time stability and robust control systhesis of uncertain switched systems, SIAM, J. Control Optim., 43 (2004/05), 1253-1271. doi: 10.1137/S0363012903425593. [19] A. Pazy, The Lyapunov method for semigroups of nonlinear contractions in Banach spaces, J. Anal. Math., 40 (1981), 239-262. doi: 10.1007/BF02790164. [20] A. Polyakov and A. Poznyak, Lyapunov function design for finite-time convergence analysis: "Twisting" controller for second-order sliding mode realization, Automatica J. IFAC, 45 (2009), 444-448. doi: 10.1016/j.automatica.2008.07.013. [21] P. Quittner, On the principle of linearized stability for variational inequalities, Math. Ann., 283 (1989), 257-270. doi: 10.1007/BF01446434. [22] P. Quittner, An instability criterion for variational inequalities, Nonlinear Analysis, 15 (1990), 1167-1180. doi: 10.1016/0362-546X(90)90052-I. [23] E. P. Ryan, An integral invariance principle for differential inclusions with applications in adaptive control, SIAM J. Control and Optim., 36 (1998), 960-980. doi: 10.1137/S0363012996301701. [24] R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970. [25] G. V. Smirnov, "Introduction to the Theory of Differential Inclusions," Graduate Studies in Mathematics, 41, American Mathematical Society, Providence, RI, 2002.

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##### References:
 [1] K. Addi, S. Adly, B. Brogliato and D. Goeleven, A method using the approach of Moreau and Panagiotopoulos for the mathematical formulation of non-regular circuits in electronics, Nonlinear Analysis C: Hybrid Systems and Applications, 1 (2007), 30-43. [2] S. Adly, Attractivity theory for second order non-smooth dynamical systems with application to dry friction, Journal of Mathematical Analysis and Applications, 322 (2006), 1055-1070. doi: 10.1016/j.jmaa.2005.09.076. [3] S. Adly and D. Goeleven, A stability theory for second-order nonsmooth dynamical systems with application to friction problems, J. Maths. Pures Appl., 83 (2004), 17-51. doi: 10.1016/S0021-7824(03)00071-0. [4] J.-P. Aubin and A. Cellina, "Differential Inclusions. Set-Valued Maps and Viability Theory," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1984. [5] J. Alvarez, I. Orlov and L. Acho, An invariance principle for discontinuous dynamic systems with applications to a Coulomb friction oscillator, Journal of Dynamic Systems, Measurement, and Control, 122 (2000), 687-690. doi: 10.1115/1.1317229. [6] S. P. Bhat and D. S. Bernstein, Finite-time stability of continuous autonomous systems, SIAM J. Control Optimization, 38 (2000), 751-766. doi: 10.1137/S0363012997321358. [7] F. H. Clarke, "Optimization and Nonsmooth Analysis," Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, New York, 1983. [8] A. F. Filippov, "Differential Equations with Discontinuous Right-Hand Sides," Kluwer, Dordrecht, The Netherlands, 1988. [9] D. Goeleven, D. Motreanu and V. V. Motreanu, On the stability of stationary solutions of first order evolution variational inequalities, Adv. Nonlinear Var. Inequal., 6 (2003), 1-30. [10] D. Goeleven and B. Brogliato, Necessary conditions of asymptotic stability for unilateral dynamical systems, Nonlinear Anal., 61 (2005), 961-1004. doi: 10.1016/j.na.2005.01.037. [11] D. Goeleven and B. Brogliato, Stability and unstability matrices for linear evolution variational inequalities, submitted, 2002. [12] K. K. Hassan, "Nonlinear Systems," Prentice-Hall, Upper Saddle River, NJ, 1996. [13] V. T. Haimo, Finite time controllers, SIAM J. Control and Optimisation, 24 (1986), 760-770. [14] T. Kato, Accretive operators and nonlinear evolutions equations in Banach spaces, in "Nonlinear Functional Analysis" (Proc. Sympos. Pure Math., Vol. XVIII, Part 1, Chicago, Ill., 1968), Amer. Math. Society, Providence, RI, 1970. [15] M. Kocan and P. Soravia, Lyapunov functions for infnite-dimensional systems, J. Funct. Anal., 192 (2002), 342-–363. doi: 10.1006/jfan.2001.3910. [16] E. Moulay and W. Perruquatti, Finite time stability of differential inclusions, IMA J. Math. Control Info., 22 (2005), 465-475. doi: 10.1093/imamci/dni039. [17] E. Moulay and W. Perruquatti, Finite time stability and stabilization of a class of continuous systems, J. Math. Anal. Appli., 323 (2006), 1430-1443. doi: 10.1016/j.jmaa.2005.11.046. [18] Y. Orlov, Finite time stability and robust control systhesis of uncertain switched systems, SIAM, J. Control Optim., 43 (2004/05), 1253-1271. doi: 10.1137/S0363012903425593. [19] A. Pazy, The Lyapunov method for semigroups of nonlinear contractions in Banach spaces, J. Anal. Math., 40 (1981), 239-262. doi: 10.1007/BF02790164. [20] A. Polyakov and A. Poznyak, Lyapunov function design for finite-time convergence analysis: "Twisting" controller for second-order sliding mode realization, Automatica J. IFAC, 45 (2009), 444-448. doi: 10.1016/j.automatica.2008.07.013. [21] P. Quittner, On the principle of linearized stability for variational inequalities, Math. Ann., 283 (1989), 257-270. doi: 10.1007/BF01446434. [22] P. Quittner, An instability criterion for variational inequalities, Nonlinear Analysis, 15 (1990), 1167-1180. doi: 10.1016/0362-546X(90)90052-I. [23] E. P. Ryan, An integral invariance principle for differential inclusions with applications in adaptive control, SIAM J. Control and Optim., 36 (1998), 960-980. doi: 10.1137/S0363012996301701. [24] R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970. [25] G. V. Smirnov, "Introduction to the Theory of Differential Inclusions," Graduate Studies in Mathematics, 41, American Mathematical Society, Providence, RI, 2002.
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