Advanced Search
Article Contents
Article Contents

Non-convex sweeping processes in the space of regulated functions

  • *Corresponding author

    *Corresponding author

This work is dedicated to the memory of Jaroslav Kurzweil

Supported by the GAČR Grant No. 20-14736S, RVO: 67985840, and by the European Regional Development Fund, Project No. CZ.02.1.01/0.0/0.0/16_019/0000778. The third author is a member of GNAMPA-INdAM

Abstract Full Text(HTML) Figure(1) Related Papers Cited by
  • The aim of this paper is to study a wide class of non-convex sweeping processes with moving constraint whose translation and deformation are represented by regulated functions, i. e., functions of not necessarily bounded variation admitting one-sided limits at every point. Assuming that the time-dependent constraint is uniformly prox-regular and has uniformly non-empty interior, we prove existence and uniqueness of solutions, as well as continuous data dependence with respect to the sup-norm.

    Mathematics Subject Classification: 34G25, 34A60, 47J20, 49J52, 74C05.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Violation of the uniform non-empty interior condition

  • [1] S. AdlyF. Nacry and L. Thibault, Discontinuous sweeping process with prox-regular sets, ESAIM: COCV, 23 (2017), 1293-1329.  doi: 10.1051/cocv/2016053.
    [2] H. Benabdellah, Existence of solutions to the nonconvex sweeping process, J. Differ. Equ., 164 (2000), 286-295.  doi: 10.1006/jdeq.1999.3756.
    [3] F. Bernicot and J. Venel, Existence of solutions for second-order differential inclusions involving proximal normal cones, J. Math. Pures Appl., 98 (2012), 257-294.  doi: 10.1016/j.matpur.2012.05.001.
    [4] F. Bernicot and J. Venel, Sweeping process by prox-regular sets in Riemannian Hilbert manifolds, J. Differ. Equ., 259 (2015), 4086-4121.  doi: 10.1016/j.jde.2015.05.011.
    [5] M. Brokate and P. Krejčí, Duality in the space of regulated functions and the play operator, Math. Zeit., 245 (2003), 667-688.  doi: 10.1007/s00209-003-0563-6.
    [6] C. Castaing, Sur une nouvelle classe d'équation d'évolution dans les espaces de Hilbert, Sém. Anal. Conv., 13 (1983), 28 pp.
    [7] N. Chemetov and M. D. P. Monteiro Marques, Non-convex quasi-variational differential inclusions, Set-Valued Anal., 15 (2007), 209-221.  doi: 10.1007/s11228-007-0045-9.
    [8] F. H. ClarkeR. J. Stern and P. R. Wolenski, Proximal smoothness and the lower-$C^2$ property., J. Convex Anal., 2 (1995), 117-144. 
    [9] F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, New York, 1998.
    [10] G. Colombo and V. V. Goncharov, The sweeping process without convexity, Set-Valued Anal., 7 (1999), 357-374.  doi: 10.1023/A:1008774529556.
    [11] G. Colombo and M. D. P. Monteiro Marques, Sweeping by a continuous prox-regular set, J. Differ. Equ., 187 (2003), 46-62.  doi: 10.1016/S0022-0396(02)00021-9.
    [12] G. Colombo and L. Thibault, Prox-regular sets and applications, in Handbook of Nonconvex Analysis and Applications, Int. Press, Somerville, MA, (2010), 99–182.
    [13] J. F. Edmond and L. Thibault, BV solutions of nonconvex sweeping process differential inclusions with perturbation, J. Differ. Equ., 226 (2006), 135-179.  doi: 10.1016/j.jde.2005.12.005.
    [14] H. Federer, Curvature measures, Trans. Amer. Math. Soc., 93 (1959), 418-491.  doi: 10.2307/1993504.
    [15] D. Fraňková, Regulated functions with values in Banach space, Math. Bohem., 144 (2019), 437-456.  doi: 10.21136/MB.2019.0124-19.
    [16] C. S. Hönig, Volterra Stieltjes-Integral Equations, North Holland and American Elsevier, Amsterdam and New York, 1975.
    [17] M. A. Krasnosel'skiǐ and A. V. Pokrovskiǐ, Systems with Hysteresis, Springer-Verlag, Berlin Heidelberg, 1989.
    [18] P. Krejčí, Vector hysteresis models, Euro. Jnl. Appl. Math., 2 (1991), 281-292.  doi: 10.1017/S0956792500000541.
    [19] P. Krejčí, Hysteresis, Convexity, and Dissipation in Hyperbolic Equations, Gakkōtosho, Tokyo, 1996.
    [20] P. Krejčí, The Kurzweil integral with exclusion of negligible sets, Math. Bohem., 128 (2003), 277-292. 
    [21] P. Krejčí, The Kurzweil integral and hysteresis, J. Phys.:Conference Series, 55 (2006), 144-154. 
    [22] P. Krejčí and P. Laurençot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183. 
    [23] P. KrejčíG. A. Monteiro and V. Recupero, Explicit and implicit non-convex sweeping processes in the space of absolutely continuous functions, Appl. Math. Optim., 84 (2021), 1477-1504.  doi: 10.1007/s00245-021-09801-8.
    [24] J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czech. Math. J., 7 (1957), 418-449. 
    [25] J. Kurzweil, Generalized Ordinary Differential Equations, World Scientific Publishing, Hackensack, 2012. doi: 10.1142/9789814324038.
    [26] G. A. Monteiro and A. Slavík, Generalized elementary functions, J. Math. Anal. Appl., 411 (2014), 828-852.  doi: 10.1016/j.jmaa.2013.10.010.
    [27] G. A. Monteiro, A. Slavík and M. Tvrdý, Kurzweil–Stieltjes Integral: Theory and Applications, World Scientific Publishing, New Jersey, 2019.
    [28] M. D. P. Monteiro Marques, Perturbations convexes semi-continues supérieurement de problèmes d'évolution dans les espaces de Hilbert, Sém. Anal. Conv., 14 (1984), 23 pp.
    [29] M. D. P. Monteiro Marques, Rafle par un convexe continu d'intérieur non vide en dimension infinie, Sém. Anal. Conv., 16 (1986), 11 pp.
    [30] M. D. P. Monteiro Marques, Differential Inclusions in Nonsmooth Mechanical Problems - Shocks and Dry Friction, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-7614-8.
    [31] J. J. Moreau, Rafle par un convexe variable I, Sém. Anal. Conv., 1 (1971), 43 pp.
    [32] J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differ. Equ., 26 (1977), 347-374.  doi: 10.1016/0022-0396(77)90085-7.
    [33] F. Nacry and L. Thibault, BV prox-regular sweeping process with bounded truncated variation, Optimization, 69 (2020), 1391-1437.  doi: 10.1080/02331934.2018.1514039.
    [34] J. Noel and L. Thibault, Nonconvex sweeping process with a moving set depending on the state, Vietnam J. Math., 42 (2014), 595-612.  doi: 10.1007/s10013-014-0109-8.
    [35] K. Nyström and T. Önskog, Remarks on the Skorohod problem and reflected Lévy driven SDEs in time-dependent domains, Stochastics, 87 (2015), 747-765.  doi: 10.1080/17442508.2014.1000327.
    [36] R. A. PoliquinR. T. Rockafellar and L. Thibault, Local differentiability of distance functions, Trans. Amer. Math. Soc., 352 (2000), 5231-5249.  doi: 10.1090/S0002-9947-00-02550-2.
    [37] V. Recupero, $BV$ solutions of rate independent variational inequalities, Ann. Sc. Norm. Super. Pisa Cl. Sc., 10 (2011), 269-315. 
    [38] V. Recupero and F. Santambrogio, Sweeping processes with prescribed behavior on jumps, Ann. Mat. Pura Appl., 197 (2018), 1311-1332.  doi: 10.1007/s10231-018-0726-z.
    [39] L. Thibault, Sweeping process with regular and nonregular sets, J. Differ. Equ., 193 (2003), 1-26.  doi: 10.1016/S0022-0396(03)00129-3.
    [40] L. Thibault, Moreau sweeping process with bounded truncated retraction, J. Convex Anal., 23 (2016), 1051-1098. 
    [41] M. Valadier, Quelques problèmes d'entrainement unilatéral en dimension finie, Sém. Anal. Conv., 18 (1988), 21 pp.
    [42] J. Venel, A numerical scheme for a class of sweeping processes, Numer. Math., 118 (2011), 367-400.  doi: 10.1007/s00211-010-0329-0.
    [43] J. P. Vial, Strong and weak convexity of sets and functions, Math. Oper. Res., 8 (1983), 231-259.  doi: 10.1287/moor.8.2.231.
  • 加载中



Article Metrics

HTML views(475) PDF downloads(156) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint