Sweeping by a tame process
Annales de l'Institut Fourier, Volume 67 (2017) no. 5, p. 2201-2223
We show that any semi-algebraic sweeping process admits piecewise absolutely continuous solutions (trajectories), and any such bounded trajectory must have finite length. Analogous results hold more generally for sweeping processes definable in o-minimal structures. This extends previous work on (sub)gradient dynamical systems beyond monotone sweeping sets.
Nous montrons l’existence des solutions (orbites) absolument continues par morceaux pour le processus de rafle défini par un opérateur multivoque semi-algébrique (ou plus généralement, o-minimal). Nous établissons que de telles orbites bornées sont de longueur finie. Cette contribution, dans le cas particulier où le processus de rafle correspond aux sous-niveaux d’une fonction (non nécessairement régulière), généralise les résultats connus pour les orbites des systèmes dynamiques de type sous-gradient.
Received : 2015-09-14
Revised : 2016-09-29
Accepted : 2016-10-27
Published online : 2017-11-17
DOI : https://doi.org/10.5802/aif.3133
Classification:  34A26,  34A60,  49J53,  14P10
Keywords: Sweeping process, semialgebraic, o-minimal, desingularization, subgradient
@article{AIF_2017__67_5_2201_0,
     author = {Daniilidis, Aris and Drusvyatskiy, Dmitriy},
     title = {Sweeping by a tame process},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {5},
     year = {2017},
     pages = {2201-2223},
     doi = {10.5802/aif.3133},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2017__67_5_2201_0}
}
Sweeping by a tame process. Annales de l'Institut Fourier, Volume 67 (2017) no. 5, pp. 2201-2223. doi : 10.5802/aif.3133. https://aif.centre-mersenne.org/item/AIF_2017__67_5_2201_0/

[1] Benabdellah, Houcine Existence of solutions to the nonconvex sweeping process, J. Differ. Equations, Tome 164 (2000) no. 2, pp. 286-295 | Article | Zbl 0957.34061

[2] Bolte, Jérôme; Daniilidis, Aris; Lewis, Adrian S. The Łojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems, SIAM J. Optim., Tome 17 (2007) no. 4, pp. 1205-1223 | Article | Zbl 1129.26012

[3] Bolte, Jérôme; Daniilidis, Aris; Lewis, Adrian S.; Shiota, Masahiro Clarke subgradients of stratifiable functions, SIAM J. Optim., Tome 18 (2007) no. 2, pp. 556-572 | Article | Zbl 1142.49006

[4] Bolte, Jérôme; Daniilidis, Aris; Ley, Olivier; Mazet, Laurent Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity, Trans. Am. Math. Soc., Tome 362 (2010) no. 6, pp. 3319-3363 | Article | Zbl 1202.26026

[5] Castaing, Charles; Monteiro Marques, Manuel Evolution problems associated with nonconvex closed moving sets with bounded variation, Port. Math., Tome 53 (1996) no. 1, pp. 73-87 | Zbl 0848.35052

[6] Colombo, Giovanni; Goncharov, Vladimir V. The sweeping processes without convexity, Set-Valued Var. Anal., Tome 7 (1999) no. 4, pp. 357-374 | Article | Zbl 0957.34060

[7] Colombo, Giovanni; Henrion, René; Hoang, N. D.; Mordukhovich, Boris S. Optimal control of the sweeping process, Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms, Tome 19 (2012) no. 1-2, pp. 117-159 | Zbl 1264.49013

[8] Colombo, Giovanni; Henrion, René; Hoang, N. D.; Mordukhovich, Boris S. Discrete approximations of a controlled sweeping process, Set-Valued Var. Anal., Tome 23 (2015) no. 1, pp. 69-86 | Article | Zbl 1312.49015

[9] Colombo, Giovanni; Monteiro Marques, Manuel Sweeping by a continuous prox-regular set, J. Differ. Equations, Tome 187 (2003) no. 1, pp. 46-62 | Article | Zbl 1029.34052

[10] Daniilidis, Aris; David, Guy; Durand-Cartagena, Estibalitz; Lemenant, Antoine Rectifiability of self-contracted curves in the Euclidean space and applications, J. Geom. Anal., Tome 25 (2015) no. 2, pp. 1211-1239 | Article | Zbl 1326.53009

[11] Daniilidis, Aris; Drusvyatskiy, Dmitriy; Lewis, Adrian S. Orbits of geometric descent, Can. Math. Bull., Tome 58 (2015) no. 1, pp. 44-50 | Article | Zbl 06418022

[12] Daniilidis, Aris; Ley, Olivier; Sabourau, Stéphane Asymptotic behaviour of self-contracted planar curves and gradient orbits of convex functions, J. Math. Pures Appl., Tome 94 (2010) no. 2, pp. 183-199 | Article | Zbl 1201.37023

[13] Daniilidis, Aris; Pang, Jeffrey C.H. Continuity and differentiability of set-valued maps revisited in the light of tame geometry, J. Lond. Math. Soc., Tome 83 (2011) no. 3, pp. 637-658 | Article | Zbl 1214.49016

[14] Van Den Dries, Lou; Miller, Chris Geometric categories and o-minimal structures, Duke Math. J., Tome 84 (1996) no. 2, pp. 497-540 | Article | Zbl 0889.03025

[15] Drusvyatskiy, Dmitriy; Lewis, Adrian S. Semi-algebraic functions have small subdifferentials, Math. Program., Tome 140 (2013) no. 1, pp. 5-29 | Article | Zbl 1270.49013

[16] Edmond, Jean Fenel; Thibault, Lionel BV solutions of nonconvex sweeping process differential inclusion with perturbation, J. Differ. Equations, Tome 226 (2006) no. 1, pp. 135-179 | Article | Zbl 1110.34038

[17] Georgiev, Nadezhda Bogdan; Ribarska On sweeping process with the cone of limiting normals, Set-Valued Var. Anal., Tome 21 (2013) no. 4, pp. 673-689 | Article | Zbl 1287.34008

[18] Ioffe, Alexander D. Metric regularity and subdifferential calculus, Russ. Math. Surv., Tome 55 (2000) no. 3, pp. 501-558 | Article | Zbl 0979.49017

[19] Ioffe, Alexander D. Critical values of set-valued maps with stratifiable graphs. Extensions of Sard and Smale-Sard theorems, Proc. Am. Math. Soc., Tome 136 (2008) no. 9, pp. 3111-3119 | Article | Zbl 1191.49015

[20] Ioffe, Alexander D. An invitation to tame optimization, SIAM J. Optim., Tome 19 (2009) no. 4, pp. 1894-1917 | Article | Zbl 1182.90083

[21] Kunze, Markus; Monteiro Marques, Manuel On parabolic quasi-variational inequalities and state-dependent sweeping processes, Topol. Methods Nonlinear Anal., Tome 12 (1998) no. 1, pp. 179-191 | Article | Zbl 0923.34018

[22] Kunze, Markus; Monteiro Marques, Manuel Degenerate sweeping processes, Variations of domain and free-boundary problems in solid mechanics (Paris, 1997), Kluwer Acad. Publ. (Solid Mech. Appl.) Tome 66 (1999), pp. 301-307

[23] Kunze, Markus; Monteiro Marques, Manuel An introduction to Moreau’s sweeping process, Impacts in mechanical systems (Grenoble, 1999), Springer (Lecture Notes in Physics) Tome 551 (2000), pp. 1-60 | Zbl 1047.34012

[24] Kurdyka, Krzysztof On gradients of functions definable in o-minimal structures, Ann. Inst. Fourier, Tome 48 (1998) no. 3, pp. 769-783 | Article | Zbl 0934.32009

[25] Longinetti, Marco; Manselli, Paolo; Venturi, Adriana On steepest descent curves for quasi convex families in n , Math. Nachr., Tome 288 (2015) no. 4, pp. 420-442 | Article | Zbl 1318.52004

[26] Manselli, Paolo; Pucci, Carlo Maximum length of steepest descent curves for quasi-convex functions, Geom. Dedicata, Tome 38 (1991) no. 2, pp. 211-227 | Article | Zbl 0724.52006

[27] Mordukhovich, Boris S. Variational analysis and generalized differentiation I & II, Springer, Grundlehren der Mathematischen Wissenschaften, Tome 331/332 (2006), xxii+579/xxii+610 pages | Zbl 1100.49002

[28] Moreau, Jean Jacques Evolution problem associated with a moving convex set in a Hilbert space, J. Differ. Equations, Tome 26 (1977), pp. 347-374 | Article | Zbl 0356.34067

[29] Nguyen, Hoang Dinh Variational analysis and optimal control of the sweeping process, Wayne State University (USA) (2011) (Ph. D. Thesis)

[30] Palis, Jacob Jun.; De Melo, Welington Geometric theory of dynamical systems. An introduction, Springer (1982), xii+198 pages | Zbl 0491.58001

[31] Rockafellar, R.Tyrrell; Wets, Roger J.-B. Variational analysis, Springer, Grundlehren der Mathematischen Wissenschaften, Tome 317 (1998), xiii+733 pages | Zbl 0888.49001

[32] Thibault, Lionel Sweeping process with regular and nonregular sets, J. Differ. Equations, Tome 193 (2003) no. 1, pp. 1-26 | Article | Zbl 1037.34007