# 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:
 [1] M. Brokate, Hysteresis operators,, in Phase Transitions and Hysteresis, (1994), 1. doi: 10.1007/BFb0073394. Google Scholar [2] M. Brokate, Rate independent hysteresis,, in Lectures on applied mathematics (eds. H.-J. Bungartz, (2000), 207. Google Scholar [3] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions,, Springer, (1996). doi: 10.1007/978-1-4612-4048-8. Google Scholar [4] E. Della Torre, E. Pinzaglia and E. Cardelli, Vector modeling - part I: Generalized hysteresis model,, Phys. B, 372 (2006), 111. doi: 10.1016/j.physb.2005.10.028. Google Scholar [5] E. Della Torre, E. Pinzaglia and E. Cardelli, Vector modeling - part II: Ellipsoidal vector hysteresis model. Numerical application to a 2d case,, Phys. B, 372 (2006), 115. doi: 10.1016/j.physb.2005.10.029. Google Scholar [6] D. Ekanayake and R. Iyer, Study of a play-like operator,, Phys. B, 403 (2008), 456. doi: 10.1016/j.physb.2007.08.074. Google Scholar [7] L. Gasiński, Evolution hemivariational inequality with hysteresis operator in higher order term,, Acta Math. Sin. (Engl. Ser.), 24 (2008), 107. doi: 10.1007/s10114-007-0997-6. Google Scholar [8] M. Jais, Classical and weak solutions for semilinear parabolic equations with Preisach hysteresis,, Opuscula Math., 28 (2008), 47. Google Scholar [9] B. Kaltenbacher and M. Kaltenbacher, Modeling and iterative identification of hysteresis via Preisach operators in pdes,, in Lectures on advanced computational methods in mechanics, (2007), 1. Google Scholar [10] O. Klein, Representation of hysteresis operators acting on vector-valued monotaffine functions,, Adv. Math. Sci. Appl., 22 (2012), 471. Google Scholar [11] O. Klein, Representation of hysteresis operators for vector-valued inputs by functions on strings,, Phys. B, 407 (2012), 1399. doi: 10.1016/j.physb.2011.10.015. Google Scholar [12] O. Klein, Darstellung von Hysterese-Operatoren mit Stückweise Monotaffinen Input-Funktionen Durch Funktionen auf Strings,, (German) [Representation of hysteressi operator with piecewise monotaffine input functions by functions on strings], (2013). Google Scholar [13] O. Klein, A representation result for hysteresis operators with vector valued inputs and its application to models for magnetic materials,, Phys. B, 435 (2014), 113. doi: 10.1016/j.physb.2013.09.034. Google Scholar [14] P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, volume 8 of Gakuto Int. Series Math. Sci. & Appl., Gakkōtosho, (1996). Google Scholar [15] P. Krejčí and M. Liero, Rate independent Kurzweil processes,, Appl. Math., 54 (2009), 117. doi: 10.1007/s10492-009-0009-5. Google Scholar [16] P. Krejčí and V. Recupero, Comparing BV solutions of rate independent processes,, J. Convex Anal., 21 (2014), 121. Google Scholar [17] K. Löschner and M. Brokate, Some mathematical properties of a vector Preisach operator,, Phys. B, 403 (2008), 250. doi: 10.1016/j.physb.2007.08.021. Google Scholar [18] I. D. Mayergoyz, Mathematical Models of Hysteresis and their Applications,, 2nd edition, (2003). Google Scholar [19] M. Miettinen and P. Panagiotopoulos, Hysteresis and hemivariational inequalities: Semilinear case,, J. Global Optim., 13 (1998), 269. doi: 10.1023/A:1008288928441. Google Scholar [20] V. Recupero, BV-solutions of rate independent variational inequalities,, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 10 (2011), 269. Google Scholar [21] X. Tan, J. S. Baras and P. Krishnaprasad, Control of hysteresis in smart actuators with application to micro-positioning,, Systems & Control Letters, 54 (2005), 483. doi: 10.1016/j.sysconle.2004.09.013. Google Scholar [22] A. Visintin, Differential Models of Hysteresis, volume 111 of Applied Mathematical Sciences., Springer-Verlag, (1994). doi: 10.1007/978-3-662-11557-2. Google Scholar [23] C. Visone, Hysteresis modelling and compensation for smart sensors and actuators,, J. Phys.: Conf. Ser., 138 (2008). doi: 10.1088/1742-6596/138/1/012028. Google Scholar

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##### References:
 [1] M. Brokate, Hysteresis operators,, in Phase Transitions and Hysteresis, (1994), 1. doi: 10.1007/BFb0073394. Google Scholar [2] M. Brokate, Rate independent hysteresis,, in Lectures on applied mathematics (eds. H.-J. Bungartz, (2000), 207. Google Scholar [3] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions,, Springer, (1996). doi: 10.1007/978-1-4612-4048-8. Google Scholar [4] E. Della Torre, E. Pinzaglia and E. Cardelli, Vector modeling - part I: Generalized hysteresis model,, Phys. B, 372 (2006), 111. doi: 10.1016/j.physb.2005.10.028. Google Scholar [5] E. Della Torre, E. Pinzaglia and E. Cardelli, Vector modeling - part II: Ellipsoidal vector hysteresis model. Numerical application to a 2d case,, Phys. B, 372 (2006), 115. doi: 10.1016/j.physb.2005.10.029. Google Scholar [6] D. Ekanayake and R. Iyer, Study of a play-like operator,, Phys. B, 403 (2008), 456. doi: 10.1016/j.physb.2007.08.074. Google Scholar [7] L. Gasiński, Evolution hemivariational inequality with hysteresis operator in higher order term,, Acta Math. Sin. (Engl. Ser.), 24 (2008), 107. doi: 10.1007/s10114-007-0997-6. Google Scholar [8] M. Jais, Classical and weak solutions for semilinear parabolic equations with Preisach hysteresis,, Opuscula Math., 28 (2008), 47. Google Scholar [9] B. Kaltenbacher and M. Kaltenbacher, Modeling and iterative identification of hysteresis via Preisach operators in pdes,, in Lectures on advanced computational methods in mechanics, (2007), 1. Google Scholar [10] O. Klein, Representation of hysteresis operators acting on vector-valued monotaffine functions,, Adv. Math. Sci. Appl., 22 (2012), 471. Google Scholar [11] O. Klein, Representation of hysteresis operators for vector-valued inputs by functions on strings,, Phys. B, 407 (2012), 1399. doi: 10.1016/j.physb.2011.10.015. Google Scholar [12] O. Klein, Darstellung von Hysterese-Operatoren mit Stückweise Monotaffinen Input-Funktionen Durch Funktionen auf Strings,, (German) [Representation of hysteressi operator with piecewise monotaffine input functions by functions on strings], (2013). Google Scholar [13] O. Klein, A representation result for hysteresis operators with vector valued inputs and its application to models for magnetic materials,, Phys. B, 435 (2014), 113. doi: 10.1016/j.physb.2013.09.034. Google Scholar [14] P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, volume 8 of Gakuto Int. Series Math. Sci. & Appl., Gakkōtosho, (1996). Google Scholar [15] P. Krejčí and M. Liero, Rate independent Kurzweil processes,, Appl. Math., 54 (2009), 117. doi: 10.1007/s10492-009-0009-5. Google Scholar [16] P. Krejčí and V. Recupero, Comparing BV solutions of rate independent processes,, J. Convex Anal., 21 (2014), 121. Google Scholar [17] K. Löschner and M. Brokate, Some mathematical properties of a vector Preisach operator,, Phys. B, 403 (2008), 250. doi: 10.1016/j.physb.2007.08.021. Google Scholar [18] I. D. Mayergoyz, Mathematical Models of Hysteresis and their Applications,, 2nd edition, (2003). Google Scholar [19] M. Miettinen and P. Panagiotopoulos, Hysteresis and hemivariational inequalities: Semilinear case,, J. Global Optim., 13 (1998), 269. doi: 10.1023/A:1008288928441. Google Scholar [20] V. Recupero, BV-solutions of rate independent variational inequalities,, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 10 (2011), 269. Google Scholar [21] X. Tan, J. S. Baras and P. Krishnaprasad, Control of hysteresis in smart actuators with application to micro-positioning,, Systems & Control Letters, 54 (2005), 483. doi: 10.1016/j.sysconle.2004.09.013. Google Scholar [22] A. Visintin, Differential Models of Hysteresis, volume 111 of Applied Mathematical Sciences., Springer-Verlag, (1994). doi: 10.1007/978-3-662-11557-2. Google Scholar [23] C. Visone, Hysteresis modelling and compensation for smart sensors and actuators,, J. Phys.: Conf. Ser., 138 (2008). doi: 10.1088/1742-6596/138/1/012028. Google Scholar
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