American Institute of Mathematical Sciences

Advanced
June  2015, 35(6): 2405-2421. doi: 10.3934/dcds.2015.35.2405

Weak differentiability of scalar hysteresis operators

Martin Brokate 1, and  Pavel Krejčí 2,

1. 

Fakultät für Mathematik, TU München, Boltzmannstr. 3, D 85747 Garching bei München
2. 

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

Received  January 2014 Revised  May 2014 Published  December 2014

Rate independent evolutions can be formulated as operators, called hysteresis operators, between suitable function spaces. In this paper, we present some results concerning the existence and the form of directional derivatives and of Hadamard derivatives of such operators in the scalar case, that is, when the driving (input) function is a scalar function.
Keywords: Preisach, rate independence, evolution variational inequalities, play, Prandtl-Ishlinskii, differentiability, Hysteresis, gliding maximum., accumulated maximum.
Mathematics Subject Classification: Primary: 47J40, 47T20; Secondary: 34C55, 49J5.
Citation: Martin Brokate, Pavel Krejčí. Weak differentiability of scalar hysteresis operators. Discrete & Continuous Dynamical Systems, 2015, 35 (6) : 2405-2421. doi: 10.3934/dcds.2015.35.2405
References:
[1]

J.-J. Moreau, Problème d'evolution associé à un convexe mobile d'un espace hilbertien, C. R. Acad. Sci. Paris Sér. A-B, 276 (1973), A791-A794. (In French.)  Google Scholar

[2]

J.-J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Diff. Eq., 26 (1977), 347-374. doi: 10.1016/0022-0396(77)90085-7.  Google Scholar

[3]

A. Mielke, Evolution of rate independent systems, in Evolutionary Equations, vol. II (eds. C.M. Dafermos and and E. Feireisl), Handbook of Differential Equations, Elsevier/North Holland, Amsterdam, II (2005), 461-559.  Google Scholar

[4]

A. Mielke, F. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Ration. Mech. Anal., 162 (2002), 137-177. doi: 10.1007/s002050200194.  Google Scholar

[5]

M. A. Krasnosel'skiĭ, B. M. Darinskiĭ, I. V. Emelin, P. P. Zabrejko, E. A. Lifshits and A. V. Pokrovskiĭ, An operator-hysterant, Dokl. Akad. Nauk SSSR, 190 (1970), 34-37; Soviet Math. Dokl., 11 (1970), 29-33.  Google Scholar

[6]

M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, Systems with Hysteresis, Nauka, Moscow, 1983. (In Russian.)  Google Scholar

[7]

M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, Systems with Hysteresis, Springer, Heidelberg, 1989. doi: 10.1007/978-3-642-61302-9.  Google Scholar

[8]

A. Visintin, Differential models of hysteresis, Springer, Berlin, 1994. doi: 10.1007/978-3-662-11557-2.  Google Scholar

[9]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996. doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[10]

P. Krejĭ, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakkōtosho, Tokyo, 1996.  Google Scholar

[11]

P. Dupuis and H. Ishii, On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications, Stochastics Stochastics Rep., 35 (1991), 31-62. doi: 10.1080/17442509108833688.  Google Scholar

[12]

P. Krejčí and Ph. Laurençot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183.  Google Scholar

[13]

P. Krejčí, Hysteresis in singularly perturbed problems, in Singular Perturbations and Hysteresis (eds. M.P. Mortell, R.E. O'Malley, A. Pokrovskiĭ, and V. Sobolev), SIAM, (2005), 73-100. doi: 10.1137/1.9780898717860.ch3.  Google Scholar

[14]

M. Brokate and P. Krejčí, Duality in the space of regulated functions and the play operator, Math. Z., 245 (2003), 667-688. doi: 10.1007/s00209-003-0563-6.  Google Scholar

[15]

R. Rossi, A. Mielke and G. Savaré, A metric approach to a class of doubly nonlinear evolution equations and applications, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 7 (2008), 97-169.  Google Scholar

[16]

I. V. Girsanov, Lectures on Mathematical Theory of Extremum Problems, Springer, Berlin, 1972.  Google Scholar

[17]

D. Fraňková, Regulated functions, Math. Bohem., 116 (1991), 20-59.  Google Scholar

[18]

L. Prandtl, Ein Gedankenmodell zur kinetischen Theorie der festen Körper, Z. f. angew. Math., 8 (1928), 85-106. (In German.) Google Scholar

[19]

A. Yu. Ishlinskiĭ, Some applications of statistical methods to describing deformations of bodies, Izv. AN SSSR Techn. Ser., 9 (1928), 583-590. (In Russian.) Google Scholar

[20]

F. Preisach, Über die magnetische Nachwirkung, Z. Phys., 94 (1935), 277-302. (In German.) Google Scholar

[21]

P. Krejčí, On Maxwell equations with the Preisach operator: The one-dimensional time-periodic case, Apl. Mat., 34 (1989), 364-374.  Google Scholar

[22]

M. Brokate, Some BV properties of the Preisach hysteresis operator, Appl. Anal., 32 (1989), 229-252. doi: 10.1080/00036818908839851.  Google Scholar

show all references

References:
[1]

J.-J. Moreau, Problème d'evolution associé à un convexe mobile d'un espace hilbertien, C. R. Acad. Sci. Paris Sér. A-B, 276 (1973), A791-A794. (In French.)  Google Scholar

[2]

J.-J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Diff. Eq., 26 (1977), 347-374. doi: 10.1016/0022-0396(77)90085-7.  Google Scholar

[3]

A. Mielke, Evolution of rate independent systems, in Evolutionary Equations, vol. II (eds. C.M. Dafermos and and E. Feireisl), Handbook of Differential Equations, Elsevier/North Holland, Amsterdam, II (2005), 461-559.  Google Scholar

[4]

A. Mielke, F. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Ration. Mech. Anal., 162 (2002), 137-177. doi: 10.1007/s002050200194.  Google Scholar

[5]

M. A. Krasnosel'skiĭ, B. M. Darinskiĭ, I. V. Emelin, P. P. Zabrejko, E. A. Lifshits and A. V. Pokrovskiĭ, An operator-hysterant, Dokl. Akad. Nauk SSSR, 190 (1970), 34-37; Soviet Math. Dokl., 11 (1970), 29-33.  Google Scholar

[6]

M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, Systems with Hysteresis, Nauka, Moscow, 1983. (In Russian.)  Google Scholar

[7]

M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, Systems with Hysteresis, Springer, Heidelberg, 1989. doi: 10.1007/978-3-642-61302-9.  Google Scholar

[8]

A. Visintin, Differential models of hysteresis, Springer, Berlin, 1994. doi: 10.1007/978-3-662-11557-2.  Google Scholar

[9]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996. doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[10]

P. Krejĭ, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakkōtosho, Tokyo, 1996.  Google Scholar

[11]

P. Dupuis and H. Ishii, On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications, Stochastics Stochastics Rep., 35 (1991), 31-62. doi: 10.1080/17442509108833688.  Google Scholar

[12]

P. Krejčí and Ph. Laurençot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183.  Google Scholar

[13]

P. Krejčí, Hysteresis in singularly perturbed problems, in Singular Perturbations and Hysteresis (eds. M.P. Mortell, R.E. O'Malley, A. Pokrovskiĭ, and V. Sobolev), SIAM, (2005), 73-100. doi: 10.1137/1.9780898717860.ch3.  Google Scholar

[14]

M. Brokate and P. Krejčí, Duality in the space of regulated functions and the play operator, Math. Z., 245 (2003), 667-688. doi: 10.1007/s00209-003-0563-6.  Google Scholar

[15]

R. Rossi, A. Mielke and G. Savaré, A metric approach to a class of doubly nonlinear evolution equations and applications, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 7 (2008), 97-169.  Google Scholar

[16]

I. V. Girsanov, Lectures on Mathematical Theory of Extremum Problems, Springer, Berlin, 1972.  Google Scholar

[17]

D. Fraňková, Regulated functions, Math. Bohem., 116 (1991), 20-59.  Google Scholar

[18]

L. Prandtl, Ein Gedankenmodell zur kinetischen Theorie der festen Körper, Z. f. angew. Math., 8 (1928), 85-106. (In German.) Google Scholar

[19]

A. Yu. Ishlinskiĭ, Some applications of statistical methods to describing deformations of bodies, Izv. AN SSSR Techn. Ser., 9 (1928), 583-590. (In Russian.) Google Scholar

[20]

F. Preisach, Über die magnetische Nachwirkung, Z. Phys., 94 (1935), 277-302. (In German.) Google Scholar

[21]

P. Krejčí, On Maxwell equations with the Preisach operator: The one-dimensional time-periodic case, Apl. Mat., 34 (1989), 364-374.  Google Scholar

[22]

M. Brokate, Some BV properties of the Preisach hysteresis operator, Appl. Anal., 32 (1989), 229-252. doi: 10.1080/00036818908839851.  Google Scholar

[1]

Pavel Krejčí, Jürgen Sprekels. Clamped elastic-ideally plastic beams and Prandtl-Ishlinskii hysteresis operators. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 283-292. doi: 10.3934/dcdss.2008.1.283
[2]

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
[3]

Wei Liu, Pavel Krejčí, Guoju Ye. Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3783-3795. doi: 10.3934/dcdsb.2017190
[4]

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
[5]

Yunkyong Hyon, Do Young Kwak, Chun Liu. Energetic variational approach in complex fluids: Maximum dissipation principle. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1291-1304. doi: 10.3934/dcds.2010.26.1291
[6]

Pavel Krejčí. The Preisach hysteresis model: Error bounds for numerical identification and inversion. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 101-119. doi: 10.3934/dcdss.2013.6.101
[7]

H. O. Fattorini. The maximum principle in infinite dimension. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 557-574. doi: 10.3934/dcds.2000.6.557
[8]

Tim Alderson, Alessandro Neri. Maximum weight spectrum codes. Advances in Mathematics of Communications, 2019, 13 (1) : 101-119. doi: 10.3934/amc.2019006
[9]

Khalid Addi, Samir Adly, Hassan Saoud. Finite-time Lyapunov stability analysis of evolution variational inequalities. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1023-1038. doi: 10.3934/dcds.2011.31.1023
[10]

Vyacheslav Maksimov. The method of extremal shift in control problems for evolution variational inequalities under disturbances. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021048
[11]

Carlo Orrieri. A stochastic maximum principle with dissipativity conditions. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5499-5519. doi: 10.3934/dcds.2015.35.5499
[12]

R.L. Sheu, M.J. Ting, I.L. Wang. Maximum flow problem in the distribution network. Journal of Industrial & Management Optimization, 2006, 2 (3) : 237-254. doi: 10.3934/jimo.2006.2.237
[13]

Brian D. Ewald, Roger Témam. Maximum principles for the primitive equations of the atmosphere. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 343-362. doi: 10.3934/dcds.2001.7.343
[14]

Samir Adly, Tahar Haddad. On evolution quasi-variational inequalities and implicit state-dependent sweeping processes. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1791-1801. doi: 10.3934/dcdss.2020105
[15]

Paolo Maremonti. On the Stokes problem in exterior domains: The maximum modulus theorem. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2135-2171. doi: 10.3934/dcds.2014.34.2135
[16]

John Sheekey. A new family of linear maximum rank distance codes. Advances in Mathematics of Communications, 2016, 10 (3) : 475-488. doi: 10.3934/amc.2016019
[17]

Julia Piantadosi, Phil Howlett, Jonathan Borwein, John Henstridge. Maximum entropy methods for generating simulated rainfall. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 233-256. doi: 10.3934/naco.2012.2.233
[18]

Torsten Lindström. Discrete models and Fisher's maximum principle in ecology. Conference Publications, 2003, 2003 (Special) : 571-579. doi: 10.3934/proc.2003.2003.571
[19]

Mingshang Hu. Stochastic global maximum principle for optimization with recursive utilities. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 1-. doi: 10.1186/s41546-017-0014-7
[20]

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

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (65)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy