In this paper, we study the well−posedness (in the sense of existence and uniqueness of a solution) of a discontinuous sweeping process involving prox-regular sets in Hilbert spaces. The variation of the moving set is controlled by a positive Radon measure and the perturbation is assumed to satisfy a Lipschitz property. The existence of a solution with bounded variation is achieved thanks to the Moreau’s catching-up algorithm adapted to this kind of problem. Various properties and estimates of jumps of the solution are also provided. We give sufficient conditions to ensure the uniform prox-regularity when the moving set is described by inequality constraints. As an application, we consider a nonlinear differential complementarity system which is a combination of an ordinary differential equation with a nonlinear complementarily condition. Such problems appear in many areas such as nonsmooth mechanics, nonregular electrical circuits and control systems.
Accepté le :
DOI : 10.1051/cocv/2016053
Mots clés : Variational analysis, measure differential inclusions, sweeping process, prox-regular set, B.V. solutions, Moreau’s catching-up algorithm, nonlinear differential complementarity systems
@article{COCV_2017__23_4_1293_0,
author = {Adly, Samir and Nacry, Florent and Thibault, Lionel},
title = {Discontinuous sweeping process with prox-regular sets},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1293--1329},
publisher = {EDP-Sciences},
volume = {23},
number = {4},
year = {2017},
doi = {10.1051/cocv/2016053},
mrnumber = {3716922},
zbl = {1379.49023},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2016053/}
}
TY - JOUR AU - Adly, Samir AU - Nacry, Florent AU - Thibault, Lionel TI - Discontinuous sweeping process with prox-regular sets JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 1293 EP - 1329 VL - 23 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2016053/ DO - 10.1051/cocv/2016053 LA - en ID - COCV_2017__23_4_1293_0 ER -
%0 Journal Article %A Adly, Samir %A Nacry, Florent %A Thibault, Lionel %T Discontinuous sweeping process with prox-regular sets %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 1293-1329 %V 23 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2016053/ %R 10.1051/cocv/2016053 %G en %F COCV_2017__23_4_1293_0
Adly, Samir; Nacry, Florent; Thibault, Lionel. Discontinuous sweeping process with prox-regular sets. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 4, pp. 1293-1329. doi : 10.1051/cocv/2016053. http://www.numdam.org/articles/10.1051/cocv/2016053/
, and , Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities. Math. Program. Ser. B 148 (2014) 5–47. | DOI | MR | Zbl
, Existence of solutions to the nonconvex sweeping process. J. Differ. Eqs. 164 (2000) 286–295. | DOI | MR | Zbl
, , On various notions of regularity of sets in nonsmooth analysis. Nonlinear Anal. Ser. A: Theory Methods 48 (2002) 223–246. | DOI | MR | Zbl
and , Nonconvex sweeping process and prox-regularity in Hilbert space. J. Nonlinear Convex Anal. 6 (2005) 359–374. | MR | Zbl
C. Castaing, Equation différentielle multivoque avec contrainte sur l’état dans les espaces de Banach. Travaux Sém. Anal. Convexe. Montpellier (1978) Exposé 13. | MR
and , BV periodic solutions of an evolution problem associated with continuous moving convex sets. Set-Valued Anal. 3 (1995) 381–399. | DOI | MR | Zbl
, M.D.P. Monteiro Marques, Evolution problems associated with non-convex closed moving sets with bounded variation. Portugal. Math. 53 (1996) 73–87. | MR | Zbl
F.H. Clarke, Functional Analysis, Calculus of Variations and Optimal Control. Springer, London (2013). | MR | Zbl
and , The sweeping processes without convexity. Set-Valued Anal. 7 (1999) 357–374. | DOI | MR | Zbl
, Existence of slow solutions for a class of differential inclusions. J. Math. Anal. Appl. 96 (1983) 130–147. | DOI | MR | Zbl
N. Dinculeanu, Vector Measures, Pergamon, Oxford (1967). | MR
and , BV solutions of nonconvex sweeping process differential inclusions with perturbation. J. Differ. Eqs. 226 (2006) 135–179. | DOI | MR | Zbl
, , Slow and quasi-slow solutions of differential inclusions. Nonlinear Anal. 11 (1987) 367–377. | DOI | MR | Zbl
, Curvature measures. Trans. Amer. Math. Soc. 93 (1959) 418–491. | DOI | MR | Zbl
, An existence theorem for a class of differential equations with multivalued right-hand side. J. Math. Anal. Appl. 41 (1973) 179–186. | DOI | MR | Zbl
P. Mattila, Geometry of Sets and Measures in Euclidean Spaces. Cambridge Univ. Press, London (1995). | MR | Zbl
M.D.P. Monteiro Marques, Perturbations convexes semi-continues supérieurement de problèmes d’évolution dans les espaces de Hilbert. Travaux Sém. Anal. Convexe. Montpellier (1984) Exposé 2. | MR
and , A mathematical framework for a crowd motion model, C. R. Math. Acad. Sci. Paris 346 (2008) 1245–1250. | DOI | MR | Zbl
B.S. Mordukhovich, Variational Analysis and Generalized Differentiation I. Vol. 330 Grundlehren Series. Springer (2006). | MR | Zbl
J.J. Moreau, Rafle par un convexe variable I. Travaux Sém. Anal. Convexe. Montpellier (1971) Exposé 15. | Zbl
J.J. Moreau, On unilateral constraints, friction and plasticity. New Variational Techniques in Mathematical Physics (C.I.M.E., II Ciclo 1973). Edizioni Cremonese, Rome (1974) 171–322. | MR
J.J. Moreau, Sur les mesures différentielles des fonctions vectorielles à variation bornée. Travaux Sém. Anal. Convexe. Montpellier (1975) Exposé 17. | MR | Zbl
, Evolution problem associated with a moving convex set in a Hilbert space. J. Differ. Eqs. 26 (1977) 347–374. | DOI | MR | Zbl
J.J. Moreau, Bounded variation in time. Topics in nonsmooth mechanics, Vol. 174. Birkhäuser, Basel (1988). | MR | Zbl
, Numerical aspects of the sweeping process. Comput. Methods Appl. Mech. Engrg. 177 (1999) 329–349. | DOI | MR | Zbl
J.J. Moreau, An introduction to unilateral dynamics. Novel Approaches in Civil Engineering. Edited by M. Frémond and F. Maceri. Springer, Berlin (2002). | Zbl
and , A chain rule involving vector functions of bounded variation. J. Funct. Anal. 74 (1987) 333–345. | DOI | MR | Zbl
, , , Local differentiability of distance functions. Trans. Amer. Math. Soc. 352 (2000) 5231–5249. | DOI | MR | Zbl
R.T. Rockafellar and R.J.-B. Wets, Variational Analysis. Grundlehren der Mathematischen Wissenschaften, vol. 317. Springer, New York (1998). | MR | Zbl
, and , Stability and observer design for Lur’e systems with multivalued, nonmonotone, time-varying nonlinearities and state jumps. SIAM J. Control Optim. 52 (2014) 3639–3672. | DOI | MR | Zbl
, Sweeping process with regular and nonregular sets. J. Differ. Eqs. 193 (2003) 1–26. | DOI | MR | Zbl
M. Valadier, Quelques problèmes d’entraînement unilatéral en dimension finie, Travaux Sém. Anal. Convexe. Montpellier (1988) Expos8é. | MR | Zbl
M. Valadier, Rafle et viabilité. Travaux Sém. Anal. Convexe. Montpellier (1992) Exposé 17. | MR | Zbl
, A numerical scheme for a class of sweeping processes. Numer. Math. 118 (2011) 367–400. | DOI | MR | Zbl
, Strong and weak convexity of sets and functions. Math. Oper. Res. 8 (1983) 231–259. | DOI | MR | Zbl
Cité par Sources :
