American Institute of Mathematical Sciences

Advanced
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 - 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.   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.  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.  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], (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, 122 (2000), 687.  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.  doi: 10.1137/S0363012997321358.  Google Scholar

[7]

F. H. Clarke, "Optimization and Nonsmooth Analysis,", Canadian Mathematical Society Series of Monographs and Advanced Texts, (1983).   Google Scholar

[8]

A. F. Filippov, "Differential Equations with Discontinuous Right-Hand Sides,", Kluwer, (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.   Google Scholar

[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.  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, (1996).   Google Scholar

[13]

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

[14]

T. Kato, Accretive operators and nonlinear evolutions equations in Banach spaces,, in, (1968).   Google Scholar

[15]

M. Kocan and P. Soravia, Lyapunov functions for infnite-dimensional systems,, J. Funct. Anal., 192 (2002).  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.  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.  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, 43 (2004), 1253.  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.  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.  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.  doi: 10.1007/BF01446434.  Google Scholar

[22]

P. Quittner, An instability criterion for variational inequalities,, Nonlinear Analysis, 15 (1990), 1167.  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.  doi: 10.1137/S0363012996301701.  Google Scholar

[24]

R. T. Rockafellar, "Convex Analysis,", Princeton Mathematical Series, (1970).   Google Scholar

[25]

G. V. Smirnov, "Introduction to the Theory of Differential Inclusions,", Graduate Studies in Mathematics, 41 (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.   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.  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.  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], (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, 122 (2000), 687.  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.  doi: 10.1137/S0363012997321358.  Google Scholar

[7]

F. H. Clarke, "Optimization and Nonsmooth Analysis,", Canadian Mathematical Society Series of Monographs and Advanced Texts, (1983).   Google Scholar

[8]

A. F. Filippov, "Differential Equations with Discontinuous Right-Hand Sides,", Kluwer, (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.   Google Scholar

[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.  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, (1996).   Google Scholar

[13]

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

[14]

T. Kato, Accretive operators and nonlinear evolutions equations in Banach spaces,, in, (1968).   Google Scholar

[15]

M. Kocan and P. Soravia, Lyapunov functions for infnite-dimensional systems,, J. Funct. Anal., 192 (2002).  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.  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.  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, 43 (2004), 1253.  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.  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.  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.  doi: 10.1007/BF01446434.  Google Scholar

[22]

P. Quittner, An instability criterion for variational inequalities,, Nonlinear Analysis, 15 (1990), 1167.  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.  doi: 10.1137/S0363012996301701.  Google Scholar

[24]

R. T. Rockafellar, "Convex Analysis,", Princeton Mathematical Series, (1970).   Google Scholar

[25]

G. V. Smirnov, "Introduction to the Theory of Differential Inclusions,", Graduate Studies in Mathematics, 41 (2002).   Google Scholar

[1]

Huijuan Li, Junxia Wang. Input-to-state stability of continuous-time systems via finite-time Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019192
[2]

Volodymyr Pichkur. On practical stability of differential inclusions using Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1977-1986. doi: 10.3934/dcdsb.2017116
[3]

Shengji Li, Chunmei Liao, Minghua Li. Stability analysis of parametric variational systems. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 317-331. doi: 10.3934/naco.2011.1.317
[4]

S. J. Li, Z. M. Fang. On the stability of a dual weak vector variational inequality problem. Journal of Industrial & Management Optimization, 2008, 4 (1) : 155-165. doi: 10.3934/jimo.2008.4.155
[5]

Georges Bastin, B. Haut, Jean-Michel Coron, Brigitte d'Andréa-Novel. Lyapunov stability analysis of networks of scalar conservation laws. Networks & Heterogeneous Media, 2007, 2 (4) : 751-759. doi: 10.3934/nhm.2007.2.751
[6]

Franco Maceri, Michele Marino, Giuseppe Vairo. Equilibrium and stability of tensegrity structures: A convex analysis approach. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 461-478. doi: 10.3934/dcdss.2013.6.461
[7]

Yongchao Liu. Quantitative stability analysis of stochastic mathematical programs with vertical complementarity constraints. Numerical Algebra, Control & Optimization, 2018, 8 (4) : 451-460. doi: 10.3934/naco.2018028
[8]

Peter Giesl. Construction of a finite-time Lyapunov function by meshless collocation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2387-2412. doi: 10.3934/dcdsb.2012.17.2387
[9]

Arno Berger. On finite-time hyperbolicity. Communications on Pure & Applied Analysis, 2011, 10 (3) : 963-981. doi: 10.3934/cpaa.2011.10.963
[10]

Eugen Stumpf. Local stability analysis of differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3445-3461. doi: 10.3934/dcds.2016.36.3445
[11]

Kun Wang, Yinnian He, Yanping Lin. Long time numerical stability and asymptotic analysis for the viscoelastic Oldroyd flows. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1551-1573. doi: 10.3934/dcdsb.2012.17.1551
[12]

Arno Berger, Doan Thai Son, Stefan Siegmund. Nonautonomous finite-time dynamics. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 463-492. doi: 10.3934/dcdsb.2008.9.463
[13]

Ismail Abdulrashid, Abdallah A. M. Alsammani, Xiaoying Han. Stability analysis of a chemotherapy model with delays. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 989-1005. doi: 10.3934/dcdsb.2019002
[14]

Xing Wang, Nan-Jing Huang. Stability analysis for set-valued vector mixed variational inequalities in real reflexive Banach spaces. Journal of Industrial & Management Optimization, 2013, 9 (1) : 57-74. doi: 10.3934/jimo.2013.9.57
[15]

Sanming Liu, Zhijie Wang, Chongyang Liu. On convergence analysis of dual proximal-gradient methods with approximate gradient for a class of nonsmooth convex minimization problems. Journal of Industrial & Management Optimization, 2016, 12 (1) : 389-402. doi: 10.3934/jimo.2016.12.389
[16]

Vladimir F. Demyanov, Julia A. Ryabova. Exhausters, coexhausters and converters in nonsmooth analysis. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1273-1292. doi: 10.3934/dcds.2011.31.1273
[17]

Graeme D. Chalmers, Desmond J. Higham. Convergence and stability analysis for implicit simulations of stochastic differential equations with random jump magnitudes. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 47-64. doi: 10.3934/dcdsb.2008.9.47
[18]

Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521
[19]

Abdelhai Elazzouzi, Aziz Ouhinou. Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 115-135. doi: 10.3934/dcds.2011.30.115
[20]

Fatiha Alabau-Boussouira, Vincent Perrollaz, Lionel Rosier. Finite-time stabilization of a network of strings. Mathematical Control & Related Fields, 2015, 5 (4) : 721-742. doi: 10.3934/mcrf.2015.5.721

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences