Perturbed First Order State Dependent Moreau's Sweeping Process

Document Type : Research Paper

Authors

1 LMPA Laboratory, Department of Mathematics, Faculty of exact sciences ans computer sciences, MSBY University, Jijel, Algeria

2 LMPA Laboratory, Department of Mathematics, Faculty of Exact Sciences and Computer Science, MSBY University, Jijel, Algeria

Abstract

In this paper, we deal with the state-dependent nonconvex sweeping process motivated through quasi-variational inequalities arising in the evolution of sandpiles, quasistatic evolution problems with friction, micromechanical damage models for iron materials. We prove the existence of absolutely continuous solution for the problem in presence of a perturbation, that is an external force applied on the system. The perturbation considered here is general and take the form of a sum of a single-valued Carath'eodory mapping and a set-valued unbounded mapping.

Keywords

[1] S. Adly, F. Nacry and L. Thibault, Discontinuous sweeping process with prox-regular sets, ESAIM Control Optim.
Calc. Var. 23(4) (2017) 1293–1329.
[2] D. Affane, M. Aissous and M. F. Yarou, Existence results for sweeping process with almost convex perturbation,
Bull. Math. Soc. Sci. Math. Roumanie, 2 (2018) 119–134.
[3] D. Affane, M. Aissous and M. F. Yarou, Almost mixed semi-continuous perturbation of Moreau’s sweeping process,
Evol. Equ. Control Theory, 1 (2020), 27–38.
[4] D. Affane and M. F. Yarou, Unbounded perturbation for a class of variational inequalities, Discuss. Math. Diff.
inclu. control optim. 37 (2017) 83–99.
[5] J. P. Aubin and H. Frankowska, Set-valued analysis, Birkhauser, Boston Basel Berlin, 1990.
[6] H. Benabdellah, Existence of solutions to the nonconvex sweeping process, J. Diff. Equa. 164 (2000) 286–295.
[7] M. Bounkhel and L. Thibault, Nonconvex sweeping process and prox-regularity in Hilbert space, J. Nonlin. Convex
Anal. 6 (2) (2005) 359–374.
[8] M. Bounkhel and M. F. Yarou, Existence results for first and second order nonconvex sweeping process with delay,
Portug. Math. 61 (2) (2004) 2007–2030.
[9] B. Brogliato, The absolute stability problem and the Lagrange-Dirichlet theorem with monotone multivalued
mappings, System Contr. Letters. 51 (5) (2004) 343–353.
[10] C. Castaing, A. G. Ibrahim and M. F. Yarou, Existence problems in second order evolution inclusions: discretization and variational approach, Taiwanese J. Math. 12 (6) (2008) 1435-1477.
[11] C. Castaing, A. G. Ibrahim and M. F. Yarou, Some contributions to nonconvex sweeping process, J. Nonlin.
Convex Anal. 10 (2009) 1–20.
[12] C. Castaing, P. Raynaud de Fitte and M. Valadier, Young measures on Topological Spaces With Applications in
Control Theory and Probability Theory, Kluwer Academic Publishers, Dordrecht, 2004.
[13] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lect. Notes in Math., Springer
Verlag, Berlin 580, 1977.
[14] N. Chemetov and M. D. P. Monteiro Marques, Non-convex Quasi-variational Differential Inclusions, Set-Valued
Variat. Anal. 5 (3) (2007) 209–221.
[15] K. Chraibi, Resolution du problème de rafle et application a un problème de frottement, Topol. Metho. Nonlin.
Anal. 18 (2001) 89–102.
[16] F. H. Clarke, R. J. Stern and P. R. Wolenski, Proximal smoothness and the lower C2 property, J. Convex Anal 2
(1995) 117–144.
[17] G. Colombo and V. V. Goncharov, The sweeping processes without convexity , Set-Vaued Anal. 7 (4) (1999)
357–374.
[18] J. F. Edmond and L. Thibault, Relaxation and optimal control problem involving a perturbed sweeping process,
Math. Program. Ser. B 104 (2005) 347–373.
[19] D. Goeleven and B. Brogliato, Stability and instability matrices for linear evolution variational inequalities, IEEE
Transactions Automatic Control 49 (4) (2004) 521–534.
[20] P. Krejci and V. Recupero, Comparing BV solutions of rate independent processes, J. convex Anal. 21 (1) (2014)
121–146.
[21] M. Kunze and M. D. P. Monteiro Marques, On parabolic quasi-variational inequalities and state-dependent sweeping processes, Topol. Metho. Nonlin. Anal. 12 (1998) 179–191.
[22] M. Kunze and M. D. P. Monteiro Marques, An introduction to Moreau’s sweeping process, Lect. Notes in Physics,
Springer Verlag, Berlin, 551, 2000, 1–60.
[23] J. J. Moreau, Evolution problem asssociated with a moving convex set in a Hilbert space, J. Diff. Equ. 26 (1977)
347–374.
[24] J. Noel and L. Thibault, Nonconvex sweeping process with a moving set depending on the state, Vietnam Journ.
Math. 42 (2014) 595–612.
[25] R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions, Trans. Amer.
Math. Soc. 352 (2000) 5231–5249.
[26] G. V. Smirnov, Introduction to the theory of differential inclusions, Graduate Studies in Mathematics V. 41,
American Mathematical Society, 2002.
[27] L. Thibault, Sweeping process with regular and nonregular sets, J. Diff. Equa. 193 (2003) 1–26.
[28] E. Vilches, Inclusions différentielles sur les espaces de Hilbert avec des cones normaux à des ensembles non
réguliers, Thesis, Université de Bourgogne, France, 2017.
Volume 12, Special Issue
December 2021
Pages 605-615
  • Receive Date: 11 June 2018
  • Accept Date: 20 August 2021