doi: 10.3934/cpaa.2022087
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Non-convex sweeping processes in the space of regulated functions

1. 

Faculty of Civil Engineering, Czech Technical University, Thákurova 7, 16629 Praha 6, Czech Republic

2. 

Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 11567 Praha 1, Czech Republic

3. 

Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

*Corresponding author

This work is dedicated to the memory of Jaroslav Kurzweil

Received  November 2021 Revised  April 2022 Early access May 2022

Fund Project: 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

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.

Citation: Pavel Krejčí, Giselle Antunes Monteiro, Vincenzo Recupero. Non-convex sweeping processes in the space of regulated functions. Communications on Pure and Applied Analysis, doi: 10.3934/cpaa.2022087
References:
[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.

show all references

References:
[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.

Figure 1.  Violation of the uniform non-empty interior condition
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