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.

[2]

M. Brokate, Rate independent hysteresis,, in Lectures on applied mathematics (eds. H.-J. Bungartz, (2000), 207.

[3]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions,, Springer, (1996). doi: 10.1007/978-1-4612-4048-8.

[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.

[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.

[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.

[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.

[8]

M. Jais, Classical and weak solutions for semilinear parabolic equations with Preisach hysteresis,, Opuscula Math., 28 (2008), 47.

[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.

[10]

O. Klein, Representation of hysteresis operators acting on vector-valued monotaffine functions,, Adv. Math. Sci. Appl., 22 (2012), 471.

[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.

[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).

[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.

[14]

P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, volume 8 of Gakuto Int. Series Math. Sci. & Appl., Gakkōtosho, (1996).

[15]

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

[16]

P. Krejčí and V. Recupero, Comparing BV solutions of rate independent processes,, J. Convex Anal., 21 (2014), 121.

[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.

[18]

I. D. Mayergoyz, Mathematical Models of Hysteresis and their Applications,, 2nd edition, (2003).

[19]

M. Miettinen and P. Panagiotopoulos, Hysteresis and hemivariational inequalities: Semilinear case,, J. Global Optim., 13 (1998), 269. doi: 10.1023/A:1008288928441.

[20]

V. Recupero, BV-solutions of rate independent variational inequalities,, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 10 (2011), 269.

[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.

[22]

A. Visintin, Differential Models of Hysteresis, volume 111 of Applied Mathematical Sciences., Springer-Verlag, (1994). doi: 10.1007/978-3-662-11557-2.

[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.

show all references

References:
[1]

M. Brokate, Hysteresis operators,, in Phase Transitions and Hysteresis, (1994), 1. doi: 10.1007/BFb0073394.

[2]

M. Brokate, Rate independent hysteresis,, in Lectures on applied mathematics (eds. H.-J. Bungartz, (2000), 207.

[3]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions,, Springer, (1996). doi: 10.1007/978-1-4612-4048-8.

[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.

[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.

[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.

[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.

[8]

M. Jais, Classical and weak solutions for semilinear parabolic equations with Preisach hysteresis,, Opuscula Math., 28 (2008), 47.

[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.

[10]

O. Klein, Representation of hysteresis operators acting on vector-valued monotaffine functions,, Adv. Math. Sci. Appl., 22 (2012), 471.

[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.

[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).

[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.

[14]

P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, volume 8 of Gakuto Int. Series Math. Sci. & Appl., Gakkōtosho, (1996).

[15]

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

[16]

P. Krejčí and V. Recupero, Comparing BV solutions of rate independent processes,, J. Convex Anal., 21 (2014), 121.

[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.

[18]

I. D. Mayergoyz, Mathematical Models of Hysteresis and their Applications,, 2nd edition, (2003).

[19]

M. Miettinen and P. Panagiotopoulos, Hysteresis and hemivariational inequalities: Semilinear case,, J. Global Optim., 13 (1998), 269. doi: 10.1023/A:1008288928441.

[20]

V. Recupero, BV-solutions of rate independent variational inequalities,, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 10 (2011), 269.

[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.

[22]

A. Visintin, Differential Models of Hysteresis, volume 111 of Applied Mathematical Sciences., Springer-Verlag, (1994). doi: 10.1007/978-3-662-11557-2.

[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.

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