American Institute of Mathematical Sciences

Advanced
December  2011, 15(3): 637-650. doi: 10.3934/dcdsb.2011.15.637

Lipschitz continuous data dependence of sweeping processes in BV spaces

Pavel Krejčí 1, and  Thomas Roche 2,

1. 

Institute of Mathematics, Czech Academy of Sciences, Žitná 25, CZ-11567 Praha 1
2. 

Department of Mathematics / M6, Technische Universität München, Boltzmannstr. 3, 85748 Garching b. München, Germany

Received  February 2010 Revised  May 2010 Published  February 2011

For a rate independent sweeping process with a time dependent smooth convex constraint, we prove that the Kurzweil solution for possibly discontinuous inputs depends locally Lipschitz continuously on the data in terms of the $BV$-norm.
Keywords: Rate independence, discontinuous sweeping process, Kurzweil integral..
Mathematics Subject Classification: Primary: 49J40; Secondary: 34C5.
Citation: Pavel Krejčí, Thomas Roche. Lipschitz continuous data dependence of sweeping processes in BV spaces. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 637-650. doi: 10.3934/dcdsb.2011.15.637
References:
[1]

G. Aumann, "Reelle Funktionen,", Springer-Verlag, (1954).   Google Scholar

[2]

M. Brokate, P. Krejčí and H. Schnabel, On uniqueness in evolution quasivariational inequalities,, J. Convex Anal., 11 (2004), 111.   Google Scholar

[3]

A. L. Gurson, Continuum theory of ductile rupture by void nucleation and growth: Part I - yield criteria and flow rules for porous ductile media,, J. Engrg. Mater. Tech., 99 (1977), 2.  doi: 10.1115/1.3443401.  Google Scholar

[4]

P. Krejčí and P. Laurençot, Generalized variational inequalities,, J. Convex Anal., 9 (2002), 159.   Google Scholar

[5]

P. Krejčí and M. Liero, Rate independent Kurzweil processes,, Appl. Math., 54 (2009), 117.  doi: 10.1007/s10492-009-0009-5.  Google Scholar

[6]

J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter,, Czechoslovak Math. J., 7(82) (1957), 418.   Google Scholar

[7]

A. Mielke and F. Theil, On rate-independent hysteresis models,, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151.   Google Scholar

[8]

J.-J. Moreau, Problème d'évolution associé à un convexe mobile d'un espace hilbertien,, C. R. Acad. Sci. Paris Sér. A-B, 276 (1973).   Google Scholar

[9]

J.-J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space,, J. Differential Eq., 26 (1977), 347.  doi: 10.1016/0022-0396(77)90085-7.  Google Scholar

[10]

V. Recupero, On locally isotone rate independent operators,, Applied Mathematics Letters, 20 (2007), 1156.  doi: 10.1016/j.aml.2006.10.006.  Google Scholar

[11]

V. Recupero, $BV$-extension of rate independent operators,, Math. Nachr., 282 (2009), 86.  doi: 10.1002/mana.200610723.  Google Scholar

[12]

V. Recupero, $BV$-solutions of rate independent variational inequalities,, To appear in Ann. Sc. Norm. Super. Pisa Cl. Sci., ().   Google Scholar

[13]

U. Stefanelli, A variational characterization of rate-independent evolution,, Math. Nachr., 282 (2009), 1492.  doi: 10.1002/mana.200810803.  Google Scholar

[14]

M. Tvrdý, Regulated functions and the Perron-Stieltjes integral,, Čas. pěst. Mat., 114 (1989), 187.   Google Scholar

show all references

References:
[1]

G. Aumann, "Reelle Funktionen,", Springer-Verlag, (1954).   Google Scholar

[2]

M. Brokate, P. Krejčí and H. Schnabel, On uniqueness in evolution quasivariational inequalities,, J. Convex Anal., 11 (2004), 111.   Google Scholar

[3]

A. L. Gurson, Continuum theory of ductile rupture by void nucleation and growth: Part I - yield criteria and flow rules for porous ductile media,, J. Engrg. Mater. Tech., 99 (1977), 2.  doi: 10.1115/1.3443401.  Google Scholar

[4]

P. Krejčí and P. Laurençot, Generalized variational inequalities,, J. Convex Anal., 9 (2002), 159.   Google Scholar

[5]

P. Krejčí and M. Liero, Rate independent Kurzweil processes,, Appl. Math., 54 (2009), 117.  doi: 10.1007/s10492-009-0009-5.  Google Scholar

[6]

J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter,, Czechoslovak Math. J., 7(82) (1957), 418.   Google Scholar

[7]

A. Mielke and F. Theil, On rate-independent hysteresis models,, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151.   Google Scholar

[8]

J.-J. Moreau, Problème d'évolution associé à un convexe mobile d'un espace hilbertien,, C. R. Acad. Sci. Paris Sér. A-B, 276 (1973).   Google Scholar

[9]

J.-J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space,, J. Differential Eq., 26 (1977), 347.  doi: 10.1016/0022-0396(77)90085-7.  Google Scholar

[10]

V. Recupero, On locally isotone rate independent operators,, Applied Mathematics Letters, 20 (2007), 1156.  doi: 10.1016/j.aml.2006.10.006.  Google Scholar

[11]

V. Recupero, $BV$-extension of rate independent operators,, Math. Nachr., 282 (2009), 86.  doi: 10.1002/mana.200610723.  Google Scholar

[12]

V. Recupero, $BV$-solutions of rate independent variational inequalities,, To appear in Ann. Sc. Norm. Super. Pisa Cl. Sci., ().   Google Scholar

[13]

U. Stefanelli, A variational characterization of rate-independent evolution,, Math. Nachr., 282 (2009), 1492.  doi: 10.1002/mana.200810803.  Google Scholar

[14]

M. Tvrdý, Regulated functions and the Perron-Stieltjes integral,, Čas. pěst. Mat., 114 (1989), 187.   Google Scholar

[1]

Pavel Krejčí, Harbir Lamba, Sergey Melnik, Dmitrii Rachinskii. Kurzweil integral representation of interacting Prandtl-Ishlinskii operators. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2949-2965. doi: 10.3934/dcdsb.2015.20.2949
[2]

Dalila Azzam-Laouir, Fatiha Selamnia. On state-dependent sweeping process in Banach spaces. Evolution Equations & Control Theory, 2018, 7 (2) : 183-196. doi: 10.3934/eect.2018009
[3]

Tan H. Cao, Boris S. Mordukhovich. Optimal control of a perturbed sweeping process via discrete approximations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3331-3358. doi: 10.3934/dcdsb.2016100
[4]

Tan H. Cao, Boris S. Mordukhovich. Optimality conditions for a controlled sweeping process with applications to the crowd motion model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 267-306. doi: 10.3934/dcdsb.2017014
[5]

Lukáš Adam, Jiří Outrata. On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2709-2738. doi: 10.3934/dcdsb.2014.19.2709
[6]

Dmitrii Rachinskii. On geometric conditions for reduction of the Moreau sweeping process to the Prandtl-Ishlinskii operator. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3361-3386. doi: 10.3934/dcdsb.2018246
[7]

Doria Affane, Meriem Aissous, Mustapha Fateh Yarou. Almost mixed semi-continuous perturbation of Moreau's sweeping process. Evolution Equations & Control Theory, 2020, 9 (1) : 27-38. doi: 10.3934/eect.2020015
[8]

Tan H. Cao, Boris S. Mordukhovich. Applications of optimal control of a nonconvex sweeping process to optimization of the planar crowd motion model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4191-4216. doi: 10.3934/dcdsb.2019078
[9]

Linlin Tian, Xiaoyi Zhang, Yizhou Bai. Optimal dividend of compound poisson process under a stochastic interest rate. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-17. doi: 10.3934/jimo.2019047
[10]

Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401
[11]

Jörg Schmeling. A notion of independence via moving targets. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 269-280. doi: 10.3934/dcds.2006.15.269
[12]

Alexander Vladimirov. Equicontinuous sweeping processes. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 565-573. doi: 10.3934/dcdsb.2013.18.565
[13]

Fryderyk Falniowski, Marcin Kulczycki, Dominik Kwietniak, Jian Li. Two results on entropy, chaos and independence in symbolic dynamics. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3487-3505. doi: 10.3934/dcdsb.2015.20.3487
[14]

Alexander Alekseenko, Jeffrey Limbacher. Evaluating high order discontinuous Galerkin discretization of the Boltzmann collision integral in $ \mathcal{O}(N^2) $ operations using the discrete fourier transform. Kinetic & Related Models, 2019, 12 (4) : 703-726. doi: 10.3934/krm.2019027
[15]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212
[16]

Huijie Qiao, Jiang-Lun Wu. On the path-independence of the Girsanov transformation for stochastic evolution equations with jumps in Hilbert spaces. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1449-1467. doi: 10.3934/dcdsb.2018215
[17]

Ummugul Bulut, Edward J. Allen. Derivation of SDES for a macroevolutionary process. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1777-1792. doi: 10.3934/dcdsb.2013.18.1777
[18]

Gian-Italo Bischi, Laura Gardini, Fabio Tramontana. Bifurcation curves in discontinuous maps. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 249-267. doi: 10.3934/dcdsb.2010.13.249
[19]

G. Bonanno, Salvatore A. Marano. Highly discontinuous elliptic problems. Conference Publications, 1998, 1998 (Special) : 118-123. doi: 10.3934/proc.1998.1998.118
[20]

Mauro Garavello, Roberto Natalini, Benedetto Piccoli, Andrea Terracina. Conservation laws with discontinuous flux. Networks & Heterogeneous Media, 2007, 2 (1) : 159-179. doi: 10.3934/nhm.2007.2.159

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences