no. 5
Sweeping by a tame process
Daniilidis, Aris1; Drusvyatskiy, Dmitriy2
1 DIM–CMM, UMI CNRS 2807 Beauchef 851 (Torre Norte, piso 5), Universidad de Chile (Chile)
2 University of Washington Department of Mathematics C-138 Padelford, Seattle, WA 98195 (USA)
Annales de l'Institut Fourier, Volume 67 (2017) no. 5, pp. 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:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3133
Classification: 34A26, 34A60, 49J53, 14P10
Keywords: Sweeping process, semialgebraic, o-minimal, desingularization, subgradient
Mot clés : Processus de rafle, semi-algébrique, o-minimal, désingularisation, sous-gradient.
Daniilidis, Aris 1; Drusvyatskiy, Dmitriy 2

1 DIM–CMM, UMI CNRS 2807 Beauchef 851 (Torre Norte, piso 5), Universidad de Chile (Chile)
2 University of Washington Department of Mathematics C-138 Padelford, Seattle, WA 98195 (USA)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@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},
     pages = {2201--2223},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {67},
     number = {5},
     year = {2017},
     doi = {10.5802/aif.3133},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3133/}
}
TY  - JOUR
AU  - Daniilidis, Aris
AU  - Drusvyatskiy, Dmitriy
TI  - Sweeping by a tame process
JO  - Annales de l'Institut Fourier
PY  - 2017
SP  - 2201
EP  - 2223
VL  - 67
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3133/
DO  - 10.5802/aif.3133
LA  - en
ID  - AIF_2017__67_5_2201_0
ER  - 
%0 Journal Article
%A Daniilidis, Aris
%A Drusvyatskiy, Dmitriy
%T Sweeping by a tame process
%J Annales de l'Institut Fourier
%D 2017
%P 2201-2223
%V 67
%N 5
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3133/
%R 10.5802/aif.3133
%G en
%F AIF_2017__67_5_2201_0
Daniilidis, Aris; Drusvyatskiy, Dmitriy. 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/articles/10.5802/aif.3133/

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

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

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

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

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

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

[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, Volume 19 (2012) no. 1-2, pp. 117-159 | Zbl

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

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

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

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

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

[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., Volume 83 (2011) no. 3, pp. 637-658 | DOI | Zbl

[14] van den Dries, Lou; Miller, Chris Geometric categories and o-minimal structures, Duke Math. J., Volume 84 (1996) no. 2, pp. 497-540 | DOI | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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

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

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

Cited by Sources: