Finite-time Lyapunov stability analysis of evolution variational inequalities

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

Finite-time Lyapunov stability analysis of evolution variational inequalities

Khalid Addi ^1, , Samir Adly ^2, and Hassan Saoud ^2,

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

Abstract
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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.

Keywords: Lyapunov Stability, LaSalle Invariance Principle, Convex Analysis., Complementarity Problem, Finite-Time Stability, Variational Inequality, Differential Inclusions, Nonsmooth Analysis.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

Citation: Khalid Addi, Samir Adly, Hassan Saoud. Finite-time Lyapunov stability analysis of evolution variational inequalities. Discrete & Continuous Dynamical Systems - A, 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.
[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. 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. 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], (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, 122 (2000), 687. 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. doi: 10.1137/S0363012997321358.
[7]	F. H. Clarke, "Optimization and Nonsmooth Analysis,", Canadian Mathematical Society Series of Monographs and Advanced Texts, (1983).
[8]	A. F. Filippov, "Differential Equations with Discontinuous Right-Hand Sides,", Kluwer, (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.
[10]	D. Goeleven and B. Brogliato, Necessary conditions of asymptotic stability for unilateral dynamical systems,, Nonlinear Anal., 61 (2005), 961. 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, (1996).
[13]	V. T. Haimo, Finite time controllers,, SIAM J. Control and Optimisation, 24 (1986), 760.
[14]	T. Kato, Accretive operators and nonlinear evolutions equations in Banach spaces,, in, (1968).
[15]	M. Kocan and P. Soravia, Lyapunov functions for infnite-dimensional systems,, J. Funct. Anal., 192 (2002). 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. 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. doi: 10.1016/j.jmaa.2005.11.046.
[18]	Y. Orlov, Finite time stability and robust control systhesis of uncertain switched systems,, SIAM, 43 (2004), 1253. doi: 10.1137/S0363012903425593.
[19]	A. Pazy, The Lyapunov method for semigroups of nonlinear contractions in Banach spaces,, J. Anal. Math., 40 (1981), 239. 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. 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. doi: 10.1007/BF01446434.
[22]	P. Quittner, An instability criterion for variational inequalities,, Nonlinear Analysis, 15 (1990), 1167. 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. doi: 10.1137/S0363012996301701.
[24]	R. T. Rockafellar, "Convex Analysis,", Princeton Mathematical Series, (1970).
[25]	G. V. Smirnov, "Introduction to the Theory of Differential Inclusions,", Graduate Studies in Mathematics, 41 (2002).

show all references

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.
[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. 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. 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], (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, 122 (2000), 687. 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. doi: 10.1137/S0363012997321358.
[7]	F. H. Clarke, "Optimization and Nonsmooth Analysis,", Canadian Mathematical Society Series of Monographs and Advanced Texts, (1983).
[8]	A. F. Filippov, "Differential Equations with Discontinuous Right-Hand Sides,", Kluwer, (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.
[10]	D. Goeleven and B. Brogliato, Necessary conditions of asymptotic stability for unilateral dynamical systems,, Nonlinear Anal., 61 (2005), 961. 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, (1996).
[13]	V. T. Haimo, Finite time controllers,, SIAM J. Control and Optimisation, 24 (1986), 760.
[14]	T. Kato, Accretive operators and nonlinear evolutions equations in Banach spaces,, in, (1968).
[15]	M. Kocan and P. Soravia, Lyapunov functions for infnite-dimensional systems,, J. Funct. Anal., 192 (2002). 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. 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. doi: 10.1016/j.jmaa.2005.11.046.
[18]	Y. Orlov, Finite time stability and robust control systhesis of uncertain switched systems,, SIAM, 43 (2004), 1253. doi: 10.1137/S0363012903425593.
[19]	A. Pazy, The Lyapunov method for semigroups of nonlinear contractions in Banach spaces,, J. Anal. Math., 40 (1981), 239. 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. 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. doi: 10.1007/BF01446434.
[22]	P. Quittner, An instability criterion for variational inequalities,, Nonlinear Analysis, 15 (1990), 1167. 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. doi: 10.1137/S0363012996301701.
[24]	R. T. Rockafellar, "Convex Analysis,", Princeton Mathematical Series, (1970).
[25]	G. V. Smirnov, "Introduction to the Theory of Differential Inclusions,", Graduate Studies in Mathematics, 41 (2002).

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