zbMATH — the first resource for mathematics

Absolute stability and the Lagrange–Dirichlet theorem with monotone multivalued mappings. (English) Zbl 1157.93455
Summary: This note presents an extension of the absolute stability problem and of the Lagrange–Dirichlet theorem, when the nonlinearities entering the model are considered within the class of monotone multivalued mappings (consequently including operators with piecewise-linear graphs that may represent physical effects like Coulomb friction, dead-zones, saturations, elasto-plasticity, and unilateral constraints).

MSC:
 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 70F25 Nonholonomic systems related to the dynamics of a system of particles 70K20 Stability for nonlinear problems in mechanics
Full Text:
References:
 [1] Bainov, D.D.; Simeonov, P.S., Systems with impulse effects; stability, theory and applications, Ellis horwood series in mathematics and its applications, (1989), Wiley New York · Zbl 0676.34035 [2] Ballard, P., Formulation and well-posedness of the dynamics of rigid-body systems with perfect unilateral constraints, Phil. trans. royal soc., mathematical, physical and engineering sciences, special issue on nonsmooth mechanics, ser. A, 359, 1789, 2327-2346, (2001) · Zbl 1014.70005 [3] Brezis, H., Opérateurs maximaux monotones, north-holland mathematics studies, (1973), North-Holland Amsterdam [4] B. Brogliato, Nonsmooth Mechanics, Springer Communications and Control Engineering Series, Second Edition, Springer, London, 1999. Erratum and addenda available at http://www.inrialpes.fr/bip/people/brogliato/brogli.html [5] Brogliato, B., On the control of nonsmooth complementarity dynamical systems, Phil. trans. royal soc., mathematical, physical and engineering sciences, special issue on nonsmooth mechanics, ser. A, 359, 1789, 2369-2383, (2001) · Zbl 1014.70021 [6] Brogliato, B.; Niculescu, S.I.; Monteiro-Marques, M., On tracking control of a class of complementary-slackness hybrid mechanical systems, Systems control lett., 39, 4, 255-266, (2000) · Zbl 0951.93054 [7] Camlibel, M.K.; Heemels, W.P.M.H.; Schumacher, J.M., On linear passive complementarity systems, European J. control, 8, 3, 220-237, (2002) · Zbl 1293.93408 [8] F.H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, Wiley, New York, 1983. · Zbl 0582.49001 [9] Cottle, R.W.; Pang, J.S.; Stone, R.E., The linear complementarity problem, (1992), Academic Press New York · Zbl 0757.90078 [10] Dieudonné, J., Eléments d’analyse, tome 2, (1969), Gauthier-Villars Paris [11] Dimnet, E.; Frémond, M., Collisions involving solids and fluids, () · Zbl 1056.74548 [12] Frémond, M., Non-smooth thermomechanics, (2002), Springer Berlin · Zbl 0990.80001 [13] Glocker, Ch., Set-valued force laws, Springer lecture notes in applied mechanics, Vol. 1, (2001), Springer Berlin · Zbl 0979.70001 [14] Goeleven, D.; Motreanu, D.; Motreanu, V.V., On the stability of stationary solutions of parabolic variational inequalities, Adv. nonlinear variational inequalities, 6, 1-30, (2003) · Zbl 1085.35090 [15] Goeleven, D.; Stavroulakis, G.E.; Salmon, G.; Panagiotopoulos, P.D., Solvability theory and projection method for a class of singular variational inequalitieselastostatic unilateral contact applications, J. optim. theory appl., 95, 2, 263-293, (1997) · Zbl 0892.90180 [16] Kunze, M.; Monteiro Marques, M.D.P., An introduction to Moreau’s sweeping process, (), 1-60 · Zbl 1047.34012 [17] Lootsma, Y.J.; van der Schaft, A.J.; Camlibel, M.K., Uniqueness of solutions of linear relay systems, Automatica, 35, 3, 467-478, (1999) · Zbl 0947.93021 [18] R. Lozano, B. Brogliato, O. Egeland, B. Maschke, Dissipative Systems Analysis and Control, Springer Communications and Control Engineering Series, Springer, London, 2000. · Zbl 0958.93002 [19] Mabrouk, M., A unified variational model for the dynamics of perfect unilateral constraints, European J. mech. A/solids, 17, 5, 819-842, (1998) · Zbl 0921.70011 [20] Megretski, A.; Rantzer, A., System analysis via integral quadratic constraints, IEEE trans. automat. control, 42, 6, 819-830, (1997) · Zbl 0881.93062 [21] Michel, A.N.; Wang, K.; Hu, B., Qualitative theory of dynamical systems, (2001), Marcel Dekker New York [22] Monteiro-Marques, M.D.P., Differential inclusions in nonsmooth mechanical problems: shocks and dry friction, Pnlde, Vol. 9, (1993), Birkhäuser, Boston · Zbl 0802.73003 [23] J.J. Moreau, Fonctionnelles Convexes, Séminaire sur les équations aux dérivées partielles, Collège de France, Paris, 1966-1967. [24] J.J. Moreau, Unilateral contact and dry friction in finite freedom dynamics, in: J.J. Moreau, P.D. Panagiotopoulos (Eds.), Nonsmooth Mechanics and Applications, CISM Courses and Lectures, Vol. 302, International Centre for Mechanical Sciences, Springer, Berlin, 1988, pp. 1-82. · Zbl 0703.73070 [25] Moreau, J.J.; Valadier, M., A chain rule involving vector functions of bounded variation, J. funct. anal., 74, 333-345, (1986) · Zbl 0632.46040 [26] Panagiotopoulos, P.D., Inequality problems in mechanics and applications, convex and non-convex energy functions, (1985), Birkhäuser Boston · Zbl 0579.73014 [27] Rockafellar, R.T., Convex analysis, Princeton landmarks in mathematics, (1970), Princeton University Press Princeton, NJ · Zbl 0229.90020 [28] Rudin, W., Analyse Réelle et complexe, (1998), Dunod Paris [29] van der Schaft, A.; Schumacher, H., An introduction to hybrid dynamical systems, Springer lecture notes in control and information science, Vol. 251, (2000), Springer London · Zbl 0940.93004 [30] Vidyasagar, M., Nonlinear systems analysis, (1993), Prentice-Hall Englewood Cliffs, NJ · Zbl 0900.93132 [31] C. Canudas de Wit, B. Siciliano, G. Bastin (Eds.), Zodiac, Theory of Robot Control, Communications and Control Engineering Series, Springer, London, 1996.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.