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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

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, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. F. Filippov, "Differential Equations with Discontinuous Right-Hand Sides," Kluwer, Dordrecht, The Netherlands, 1988. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

D. Goeleven and B. Brogliato, Stability and unstability matrices for linear evolution variational inequalities, submitted, 2002. Google Scholar

[12]

K. K. Hassan, "Nonlinear Systems," Prentice-Hall, Upper Saddle River, NJ, 1996. Google Scholar

[13]

V. T. Haimo, Finite time controllers, SIAM J. Control and Optimisation, 24 (1986), 760-770.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

P. Quittner, On the principle of linearized stability for variational inequalities, Math. Ann., 283 (1989), 257-270. doi: 10.1007/BF01446434.  Google Scholar

[22]

P. Quittner, An instability criterion for variational inequalities, Nonlinear Analysis, 15 (1990), 1167-1180. doi: 10.1016/0362-546X(90)90052-I.  Google Scholar

[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.  Google Scholar

[24]

R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970.  Google Scholar

[25]

G. V. Smirnov, "Introduction to the Theory of Differential Inclusions," Graduate Studies in Mathematics, 41, American Mathematical Society, Providence, RI, 2002.  Google Scholar

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), 30-43.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. F. Filippov, "Differential Equations with Discontinuous Right-Hand Sides," Kluwer, Dordrecht, The Netherlands, 1988. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

D. Goeleven and B. Brogliato, Stability and unstability matrices for linear evolution variational inequalities, submitted, 2002. Google Scholar

[12]

K. K. Hassan, "Nonlinear Systems," Prentice-Hall, Upper Saddle River, NJ, 1996. Google Scholar

[13]

V. T. Haimo, Finite time controllers, SIAM J. Control and Optimisation, 24 (1986), 760-770.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

P. Quittner, On the principle of linearized stability for variational inequalities, Math. Ann., 283 (1989), 257-270. doi: 10.1007/BF01446434.  Google Scholar

[22]

P. Quittner, An instability criterion for variational inequalities, Nonlinear Analysis, 15 (1990), 1167-1180. doi: 10.1016/0362-546X(90)90052-I.  Google Scholar

[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.  Google Scholar

[24]

R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970.  Google Scholar

[25]

G. V. Smirnov, "Introduction to the Theory of Differential Inclusions," Graduate Studies in Mathematics, 41, American Mathematical Society, Providence, RI, 2002.  Google Scholar

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