American Institute of Mathematical Sciences

June  2015, 35(6): 2591-2614. doi: 10.3934/dcds.2015.35.2591

On the representation of hysteresis operators acting on vector-valued, left-continuous and piecewise monotaffine and continuous functions

 1 Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, 10117 Berlin

Received  December 2013 Revised  June 2014 Published  December 2014

In Brokate-Sprekels 1996, it is shown that hysteresis operators acting on scalar-valued, continuous, piecewise monotone input functions can be represented by functionals acting on alternating strings. In a number of recent papers, this representation result is extended to hysteresis operators dealing with input functions in a general topological vector space. The input functions have to be continuous and piecewise monotaffine, i.e. being piecewise the composition of two functions such that the output of a monotone increasing function is used as input for an affine function.
In the current paper, a representation result is formulated for hysteresis operators dealing with input functions being left-continuous and piecewise monotaffine and continuous. The operators are generated by functions acting on an admissible subset of the set of all strings of pairs of elements of the vector space.
Citation: Olaf Klein. On the representation of hysteresis operators acting on vector-valued, left-continuous and piecewise monotaffine and continuous functions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2591-2614. doi: 10.3934/dcds.2015.35.2591
References:

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References:
 [1] Vincenzo Recupero. Hysteresis operators in metric spaces. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 773-792. doi: 10.3934/dcdss.2015.8.773 [2] Martin Brokate, Pavel Krejčí. Weak differentiability of scalar hysteresis operators. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2405-2421. doi: 10.3934/dcds.2015.35.2405 [3] Nobuyuki Kenmochi, Jürgen Sprekels. Phase-field systems with vectorial order parameters including diffusional hysteresis effects. Communications on Pure & Applied Analysis, 2002, 1 (4) : 495-511. doi: 10.3934/cpaa.2002.1.495 [4] Rod Cross, Hugh McNamara, Leonid Kalachev, Alexei Pokrovskii. Hysteresis and post Walrasian economics. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 377-401. doi: 10.3934/dcdsb.2013.18.377 [5] J. Samuel Jiang, Hans G. Kaper, Gary K Leaf. Hysteresis in layered spring magnets. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 219-232. doi: 10.3934/dcdsb.2001.1.219 [6] 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 [7] Jana Kopfová. Nonlinear semigroup methods in problems with hysteresis. Conference Publications, 2007, 2007 (Special) : 580-589. doi: 10.3934/proc.2007.2007.580 [8] Antonio DeSimone, Natalie Grunewald, Felix Otto. A new model for contact angle hysteresis. Networks & Heterogeneous Media, 2007, 2 (2) : 211-225. doi: 10.3934/nhm.2007.2.211 [9] Pavel Gurevich. Periodic solutions of parabolic problems with hysteresis on the boundary. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1041-1083. doi: 10.3934/dcds.2011.29.1041 [10] Augusto VisintiN. On the variational representation of monotone operators. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 909-918. doi: 10.3934/dcdss.2017046 [11] Claudio Giorgi. Phase-field models for transition phenomena in materials with hysteresis. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 693-722. doi: 10.3934/dcdss.2015.8.693 [12] Emil Minchev, Mitsuharu Ôtani. $L^∞$-energy method for a parabolic system with convection and hysteresis effect. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1613-1632. doi: 10.3934/cpaa.2018077 [13] Michela Eleuteri, Pavel Krejčí. An asymptotic convergence result for a system of partial differential equations with hysteresis. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1131-1143. doi: 10.3934/cpaa.2007.6.1131 [14] Augusto Visintin. P.D.E.s with hysteresis 30 years later. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 793-816. doi: 10.3934/dcdss.2015.8.793 [15] 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 [16] Augusto Visintin. Ohm-Hall conduction in hysteresis-free ferromagnetic processes. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 551-563. doi: 10.3934/dcdsb.2013.18.551 [17] Youssef Amal, Martin Campos Pinto. Global solutions for an age-dependent model of nucleation, growth and ageing with hysteresis. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 517-535. doi: 10.3934/dcdsb.2010.13.517 [18] Dmitrii Rachinskii. Realization of arbitrary hysteresis by a low-dimensional gradient flow. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 227-243. doi: 10.3934/dcdsb.2016.21.227 [19] Takanobu Okazaki. Large time behaviour of solutions of nonlinear ode describing hysteresis. Conference Publications, 2007, 2007 (Special) : 804-813. doi: 10.3934/proc.2007.2007.804 [20] Steffen Härting, Anna Marciniak-Czochra, Izumi Takagi. Stable patterns with jump discontinuity in systems with Turing instability and hysteresis. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 757-800. doi: 10.3934/dcds.2017032

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