Abstract
The main objective of this paper is to present an overview of Moreau’s sweeping process —u’(t) ∈ N c (t) (u (t)) along with some of our results concerning new variants of this process. Several open problems are mentioned.
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C. Baiocchi and A. Capelo. Variational and Quasi-Variational Inequalities, Applications to Free Boundary Problems. John Wiley and Sons, New York, 1984.
H. Benabdellah. Existence of solutions to the nonconvex sweeping process, J. Diff. Eq. 164 (2000), 286–295.
A. Bensoussan and J.L. Lions. Impulse Control and Quasi-Variational Inequalities. Gauthier-Villars, Bordas, Paris, 1984.
M. Brokate. Elastoplastic Constitutive Laws of Nonlinear Kinematic Hardening Type. In: Brokate, M. Siddiqi, A.H. (Eds.) Functional Analysis with Current Applications in Science, Technology and Industry, Londman, Harlow (Pitman Research Lecture Notes in Mathematics), Vol. 377 (1998), 238–272.
C. Castaing and M.D.P. Monteiro Marques. Periodic Solutions of Evolution Problems Associated with a Moving Convex Set. C.R. Acad. Sci. Paris, SerA 321 (1995), 531–536.
C. Castaing and M.D.P. Monteiro Marques. BV Periodic Solutions of an Evolution Problem Associated with Continuous Moving Convex Sets. Set-valued Anal. 3 (1995), 381–399.
C. Castaing and M.D.P. Monteiro Marques. Topological Properties of Solution Sets for Sweeping Processes with Delay. Portugal Math. 54 (1997), 485–507.
G. Colombo and V. V. Goncharov. The sweeping processes without convexity. Set-valued Analysis 7(1999), 357–374.
D. Duvaut and J.L. Lions. Inequalities in Mechanics and Physics. Springer-Verlag, BerlinHeidelberg-New York, 1976.
W. Desch. Local Lipschitz continuity of the stop operator. Applications of Mathematics, 43 (1998), 461–477.
R. Glowinski, J.L. Lions and R. Tremolieres. Numerical Analysis of Variational Inequalities. North Holland Publishing Comp., Amsterdam-New York, 1981.
W. Han, B.D. Reddy and G.C. Schroeder. Qualitative and numerical analysis of quasi-static problems in elastoplasticity. SIAM J. Numer. Anal. 34 (1997), 143–177.
P. Krejčí. Hysteresis, convexity and dissipation in hyperbolic equations. Gakkotosho, Tokyo, 1996.
P. Krejčí. Evolution variational inequalities and multidimensional hysteresis operators. In: Nonlinear differential equations, Res. Notes Math. 404, Chapman & Hall CRC, Boca Raton 1999, pp. 47–110.
M. Kunze. Periodic solutions of non-linear kinematic hardening models. Math. Meth. Appl. Sci. 22 (1999), 515–529.
M. Kunze and M.D.P. Monteiro Marques. Existence of solutions for degenerate sweeping processes. J. Convex Anal. 4 (1997), 165–176.
M. Kunze and M.D.P. Monteiro Marques. On the discretization of digenerate sweeping processes. Portugal Math. 55 (1998), 219–232.
M. Kunze and M.D.P. Monteiro Marques. On parabolic quasi-variational inequalities and state-dependent sweeping processes. Topol. Methods Nonlinear Anal. 12 (1998), 179–191.
M. Kunze and M.D.P. Monteiro Marques. A note on Lipschitz continuous solutions of a parabolic quasi-variational inequality, In: Proc. Conf. Differential Equations, Macau 1998.
M. Kunze and M.D.P. Monteiro Marques. Degenerate sweeping processes. In: Argoul P., Frémond M., Nguyen Q.S. (Eds.) Proc. IUTAM Symposium on Variations of Domains and Free-Boundary Problems in Solid Mechanics, Paris 1997, Kluwer Academic Press, Dordrecht, 301–307.
M. Kunze and M.D.P. Monteiro Marques. An Introduction to Moreau’s Sweeping Process. Lecture Notes, 2000 (unpublished).
M. Kunz and J.F. Rodrigues. An elliptic quasi-variational inequality with gradient constraints and some of its applications. Math. Meth. in App. Sci. 23 (2000), 897–908.
P. Manchanda and A.H. Siddiqi. A rate-independent evolution quasi-variational inequalitis and state-dependent sweeping processes, Third World Nonlinear Analysis Conference, Catania, Italy, 19–26 July 2000.
M.D.P. Monteiro Marques. Regularization and graph approximation of a discontinuous evolution problem. J. Differential Equations 67 (1987), 145–164.
M.D.P. Monteiro Marques. Differential inclusions in nonsmooth mechanical problems-Shocks and dry friction. Birkhäuser Basel-Boston-Berlin, 1993.
J.J. Moreau. On Unilateral Constraints, Friction and Plasticity. In: Capriz G., Stampacchia G. (Eds.) New Variational Techniques in Mathematical Physics, CIME circlo Bressanone, 1973. Edizioni Cremonese, Rome, 171–322.
J.J. Moreau. Application of Convex Analysis to the Treatment of Elastoplastic Systems. In: Germain P., Nayroles B. (Eds.) Applications of Methods of Functional Analysis to Problems in Mechanics. Lecture Notes in Mathematics, Vol. 503(1976), Springer, BerlinHeidelberg-New York, 55–89.
J.J. Moreau. Evolution problem associated with a moving convex set in a Hilbert space. J. Differential Equations 26 (1977), 347–374.
J.J. Moreau. Bounded Variation in Time. In: Moreau J.J., Panagiotopoulos P.D., Strang G. (Eds.) Topics in Non-smooth Mechanics. Birkhäuser, Basel-Boston-Berlin, 1988, 1–74.
J.J. Moreau. Numerical Aspects of the Sweeping Process. Computer Methods in Applied Mechanics and Engineering 177 (1999), 329–349.
U. Mosco., Some Introductory Remarks on Implicit Variational Problems, 1–46. In: Siddiqi, A.H. (Ed.) Recent Developments in Applicable Mathematics, MacMillan India Limited, 1994.
L. Prigozhin. Variational model of sandpiles growth. European J. Appl. Math. 7 (1996), 225–235.
L. Prigozhin. On the bean critical state model in supercondctivity. European J. Appl. Math. 7 (1996), 237–247.
J.S. Raymond. A generalization of Lax-Milgram Theorem. Le Matematiche Vol. L11 (1997), 149–157.
A. H. Siddiqi and P. Manchanda. Certain remarks on a class of evolution quasi-variational inequalities. Internat. J. Math. & Math. Sc. 24 (2000), 851–855.
A.H. Siddiqi, P. Manchanda and M. Brokate. A variant of Moreau’s sweeping process, unpublished.
N.G. Yen. Linear operators satisfying the assumptions of some generalized Lax-Millgram Theorem, Third World Nonlinear Analysis Conference, Catania, Italy, 19–26 July 2000.
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Siddiqi, A.H., Manchanda, P., Brokate, M. (2002). On Some Recent Developments Concerning Moreau’s Sweeping Process. In: Siddiqi, A.H., Kočvara, M. (eds) Trends in Industrial and Applied Mathematics. Applied Optimization, vol 72. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0263-6_15
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DOI: https://doi.org/10.1007/978-1-4613-0263-6_15
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