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Ennio De Giorgi and $\mathbf\Gamma$-convergence
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 |
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. |
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. |
[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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