Weak differentiability of scalar hysteresis operators

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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

Abstract
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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 and 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.)
[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.
[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.
[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.
[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.
[6]	M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, Systems with Hysteresis, Nauka, Moscow, 1983. (In Russian.)
[7]	M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, Systems with Hysteresis, Springer, Heidelberg, 1989. doi: 10.1007/978-3-642-61302-9.
[8]	A. Visintin, Differential models of hysteresis, Springer, Berlin, 1994. doi: 10.1007/978-3-662-11557-2.
[9]	M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996. doi: 10.1007/978-1-4612-4048-8.
[10]	P. Krejĭ, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakkōtosho, Tokyo, 1996.
[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.
[12]	P. Krejčí and Ph. Laurençot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183.
[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.
[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.
[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.
[16]	I. V. Girsanov, Lectures on Mathematical Theory of Extremum Problems, Springer, Berlin, 1972.
[17]	D. Fraňková, Regulated functions, Math. Bohem., 116 (1991), 20-59.
[18]	L. Prandtl, Ein Gedankenmodell zur kinetischen Theorie der festen Körper, Z. f. angew. Math., 8 (1928), 85-106. (In German.)
[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.)
[20]	F. Preisach, Über die magnetische Nachwirkung, Z. Phys., 94 (1935), 277-302. (In German.)
[21]	P. Krejčí, On Maxwell equations with the Preisach operator: The one-dimensional time-periodic case, Apl. Mat., 34 (1989), 364-374.
[22]	M. Brokate, Some BV properties of the Preisach hysteresis operator, Appl. Anal., 32 (1989), 229-252. doi: 10.1080/00036818908839851.

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.)
[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.
[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.
[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.
[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.
[6]	M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, Systems with Hysteresis, Nauka, Moscow, 1983. (In Russian.)
[7]	M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, Systems with Hysteresis, Springer, Heidelberg, 1989. doi: 10.1007/978-3-642-61302-9.
[8]	A. Visintin, Differential models of hysteresis, Springer, Berlin, 1994. doi: 10.1007/978-3-662-11557-2.
[9]	M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996. doi: 10.1007/978-1-4612-4048-8.
[10]	P. Krejĭ, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakkōtosho, Tokyo, 1996.
[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.
[12]	P. Krejčí and Ph. Laurençot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183.
[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.
[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.
[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.
[16]	I. V. Girsanov, Lectures on Mathematical Theory of Extremum Problems, Springer, Berlin, 1972.
[17]	D. Fraňková, Regulated functions, Math. Bohem., 116 (1991), 20-59.
[18]	L. Prandtl, Ein Gedankenmodell zur kinetischen Theorie der festen Körper, Z. f. angew. Math., 8 (1928), 85-106. (In German.)
[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.)
[20]	F. Preisach, Über die magnetische Nachwirkung, Z. Phys., 94 (1935), 277-302. (In German.)
[21]	P. Krejčí, On Maxwell equations with the Preisach operator: The one-dimensional time-periodic case, Apl. Mat., 34 (1989), 364-374.
[22]	M. Brokate, Some BV properties of the Preisach hysteresis operator, Appl. Anal., 32 (1989), 229-252. doi: 10.1080/00036818908839851.

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