Kurzweil integral representation of interacting Prandtl-Ishlinskii operators

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November 2015, 20(9): 2949-2965. doi: 10.3934/dcdsb.2015.20.2949

Kurzweil integral representation of interacting Prandtl-Ishlinskii operators

Pavel Krejčí ^1, , Harbir Lamba ^2, , Sergey Melnik ^3, and Dmitrii Rachinskii ^4,

1.	Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 11567 Praha 1
2.	Department of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, VA 22030
3.	MACSI, Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland
4.	Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX 75025, United States

Received June 2014 Revised July 2015 Published September 2015

Abstract
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We consider a system of operator equations involving play and Prandtl-Ishlinskii hysteresis operators. This system generalizes the classical mechanical models of elastoplasticity, friction and fatigue by introducing coupling between the operators. We show that under quite general assumptions the coupled system is equivalent to one effective Prandtl-Ishlinskii operator or, more precisely, to a discontinuous extension of the Prandtl-Ishlinskii operator based on the Kurzweil integral of the derivative of the state function. This effective operator is described constructively in terms of the parameters of the coupled system. Our result is based on a substitution formula which we prove for the Kurzweil integral of regulated functions integrated with respect to functions of bounded variation. This formula allows us to prove the composition rule for the generalized (discontinuous) Prandtl-Ishlinskii operators. The composition rule, which underpins the analysis of the coupled model, then establishes that a composition of generalized Prandtl-Ishlinskii operators is also a generalized Prandtl-Ishlinskii operator provided that a monotonicity condition is satisfied.

Keywords: hysteresis operator, network model., composition formula, Kurzweil integral, discontinuous Prandtl-Ishlinskii operator, Regulated function, substitution formula.

Mathematics Subject Classification: Primary: 47J40; Secondary: 74C1.

Citation: Pavel Krejčí, Harbir Lamba, Sergey Melnik, Dmitrii Rachinskii. Kurzweil integral representation of interacting Prandtl-Ishlinskii operators. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 2949-2965. doi: 10.3934/dcdsb.2015.20.2949

References:

[1]	, The Science of Hysteresis (eds. I. Mayergoyz and G. Bertotti),, Elsevier, (2005).
[2]	F. Al-Bender, V. Lampaert and J. Swevers, The generalized Maxwell-slip model: A novel model for friction simulation and compensation, IEEE Trans. Automat. Control, 50 (2005), 1883-1887. doi: 10.1109/TAC.2005.858676.
[3]	G. Aumann, Reelle Funktionen, (German) [Real Functions], Springer, Berlin, 1954.
[4]	A. Barrat, M. Barthélemy and A. Vespignani, Dynamical Processes on Complex Networks, Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511791383.
[5]	G. Bertotti, V. Basso, M. LoBue and A. Magni, Thermodynamics, hysteresis, and micromagnetics, in The Science of Hysteresis (eds. I. Mayergoyz and G. Bertotti), Elsevier, (2006), Vol. II, Chapter 1, 1-106. doi: 10.1016/B978-012480874-4/50012-9.
[6]	M. Brokate, On the moving Preisach model, Math. Methods Appl. Sci., 15 (1992), 145-157. doi: 10.1002/mma.1670150302.
[7]	M. Brokate, P. Krejčí and D. Rachinskii, Some analytical properties of the multidimensional continuous Mroz model of plasticity, Control Cybernet, 27 (1998), 199-215.
[8]	M. Brokate, A. Pokrovskii and D. Rachinskii, Asymptotic stability of continuum sets of periodic solutions to systems with hysteresis, J. Math. Anal. Appl., 319 (2006), 94-109. doi: 10.1016/j.jmaa.2006.02.060.
[9]	M. Brokate, A. Pokrovskii, D. Rachinskii and O. Rasskazov, Differential equations with hysteresis via a canonical example, in The Science of Hysteresis (eds. I. Mayergoyz and G. Bertotti), Elsevier, 1 (2006), 125-291. doi: 10.1016/B978-012480874-4/50005-1.
[10]	M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, 1996. doi: 10.1007/978-1-4612-4048-8.
[11]	D. Davino, P. Krejčí and C. Visone, Fully coupled modeling of magneto-mechanical hysteresis through 'thermodynamic' compatibility, Smart Mater. Struct., 22 (2013), 095009. doi: 10.1088/0964-1726/22/9/095009.
[12]	M. A. Janaideh, S. Rakheja and C.-Y. Su, An analytical generalized Prandtl-Ishlinskii model inversion for hysteresis compensation in micropositioning control, IEEE/ASME Trans. Mechatronics, 16 (2011), 734-744. doi: 10.1109/TMECH.2010.2052366.
[13]	M. Krasnosel'skii and A. Pokrovskii, Systems with Hysteresis, Springer, 1989. doi: 10.1007/978-3-642-61302-9.
[14]	A. Krasnosel'skii and D. Rachinskii, Bifurcation of forced periodic oscillations for equations with Preisach hysteresis, J. Phys.: Conf. Ser., 22 (2005), 93-102. doi: 10.1088/1742-6596/22/1/006.
[15]	A. Krasnosel'skii and D. Rachinskii, On a bifurcation governed by hysteresis nonlinearity, NoDEA Nonlinear Differ. Equ. Appl., 9 (2002), 93-115. doi: 10.1007/s00030-002-8120-2.
[16]	P. Krejčí, The Kurzweil integral with exclusion of negligible sets, Math. Bohem., 128 (2003), 277-292.
[17]	P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakkōtosho, 1996.
[18]	P. Krejčí and K. Kuhnen, Inverse control of systems with hysteresis and creep, IEE P-Contr. Theor. Ap., 148 (2001), 185-192.
[19]	P. Krejčí and J. Kurzweil, A nonexistence result for the Kurzweil integral, Math. Bohem., 127 (2002), 571-580.
[20]	P. Krejčí, H. Lamba, S. Melnik and D. Rachinskii, Analytical solution for a class of network dynamics with mechanical and financial applications, Phys. Rev. E, 90 (2014), 032822.
[21]	P. Krejčí and Ph. Laurençot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183.
[22]	P. Krejčí, J. P. O'Kane, A. Pokrovskii and D. Rachinskii}, Properties of solutions to a class of differential models incorporating Preisach hysteresis operator, Phys. D, 241 (2012), 2010-2028. doi: 10.1016/j.physd.2011.05.005.
[23]	P. Krejčí, J. P. O'Kane, A. Pokrovskii and D. Rachinskii, Stability results for a soil model with singular hysteretic hydrology, J. Phys.: Conf. Ser. 268 (2011), 012016.
[24]	K. Kuhnen, Modeling, identification and compensation of complex hysteretic nonlinearities: A modified Prandtl-Ishlinskii approach, Eur. J. Control, 9 (2003), 407-418. doi: 10.3166/ejc.9.407-418.
[25]	J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J., 7 (1957), 418-449.
[26]	Lemaitre and J.-L. Chaboche, Mechanics of Solid Materials, Cambridge University Press, Cambridge, 1990.
[27]	I. Mayergoyz, Mathematical Models of Hysteresis and Their Applications, Elsevier, 2003.
[28]	L. Prandtl, Ein Gedankenmodell zur kinetischen Theorie der festen Körper, (German) [A model for kinetic theory of solid bodies], ZAMM Z. Ang. Math. Mech., 8 (1928), 85-106.
[29]	Š. Schwabik, On a modified sum integral of Stieltjes type, Časopis Pěst. Mat., 98 (1973), 274-277.
[30]	M. Tvrdý, Regulated functions and the Perron-Stieltjes integral, Časopis Pêst. Mat., 114 (1989), 187-209.
[31]	A. Visintin, Differential Models of Hysteresis, Springer, 1994. doi: 10.1007/978-3-662-11557-2.

show all references

References:

[1]	, The Science of Hysteresis (eds. I. Mayergoyz and G. Bertotti),, Elsevier, (2005).
[2]	F. Al-Bender, V. Lampaert and J. Swevers, The generalized Maxwell-slip model: A novel model for friction simulation and compensation, IEEE Trans. Automat. Control, 50 (2005), 1883-1887. doi: 10.1109/TAC.2005.858676.
[3]	G. Aumann, Reelle Funktionen, (German) [Real Functions], Springer, Berlin, 1954.
[4]	A. Barrat, M. Barthélemy and A. Vespignani, Dynamical Processes on Complex Networks, Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511791383.
[5]	G. Bertotti, V. Basso, M. LoBue and A. Magni, Thermodynamics, hysteresis, and micromagnetics, in The Science of Hysteresis (eds. I. Mayergoyz and G. Bertotti), Elsevier, (2006), Vol. II, Chapter 1, 1-106. doi: 10.1016/B978-012480874-4/50012-9.
[6]	M. Brokate, On the moving Preisach model, Math. Methods Appl. Sci., 15 (1992), 145-157. doi: 10.1002/mma.1670150302.
[7]	M. Brokate, P. Krejčí and D. Rachinskii, Some analytical properties of the multidimensional continuous Mroz model of plasticity, Control Cybernet, 27 (1998), 199-215.
[8]	M. Brokate, A. Pokrovskii and D. Rachinskii, Asymptotic stability of continuum sets of periodic solutions to systems with hysteresis, J. Math. Anal. Appl., 319 (2006), 94-109. doi: 10.1016/j.jmaa.2006.02.060.
[9]	M. Brokate, A. Pokrovskii, D. Rachinskii and O. Rasskazov, Differential equations with hysteresis via a canonical example, in The Science of Hysteresis (eds. I. Mayergoyz and G. Bertotti), Elsevier, 1 (2006), 125-291. doi: 10.1016/B978-012480874-4/50005-1.
[10]	M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, 1996. doi: 10.1007/978-1-4612-4048-8.
[11]	D. Davino, P. Krejčí and C. Visone, Fully coupled modeling of magneto-mechanical hysteresis through 'thermodynamic' compatibility, Smart Mater. Struct., 22 (2013), 095009. doi: 10.1088/0964-1726/22/9/095009.
[12]	M. A. Janaideh, S. Rakheja and C.-Y. Su, An analytical generalized Prandtl-Ishlinskii model inversion for hysteresis compensation in micropositioning control, IEEE/ASME Trans. Mechatronics, 16 (2011), 734-744. doi: 10.1109/TMECH.2010.2052366.
[13]	M. Krasnosel'skii and A. Pokrovskii, Systems with Hysteresis, Springer, 1989. doi: 10.1007/978-3-642-61302-9.
[14]	A. Krasnosel'skii and D. Rachinskii, Bifurcation of forced periodic oscillations for equations with Preisach hysteresis, J. Phys.: Conf. Ser., 22 (2005), 93-102. doi: 10.1088/1742-6596/22/1/006.
[15]	A. Krasnosel'skii and D. Rachinskii, On a bifurcation governed by hysteresis nonlinearity, NoDEA Nonlinear Differ. Equ. Appl., 9 (2002), 93-115. doi: 10.1007/s00030-002-8120-2.
[16]	P. Krejčí, The Kurzweil integral with exclusion of negligible sets, Math. Bohem., 128 (2003), 277-292.
[17]	P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakkōtosho, 1996.
[18]	P. Krejčí and K. Kuhnen, Inverse control of systems with hysteresis and creep, IEE P-Contr. Theor. Ap., 148 (2001), 185-192.
[19]	P. Krejčí and J. Kurzweil, A nonexistence result for the Kurzweil integral, Math. Bohem., 127 (2002), 571-580.
[20]	P. Krejčí, H. Lamba, S. Melnik and D. Rachinskii, Analytical solution for a class of network dynamics with mechanical and financial applications, Phys. Rev. E, 90 (2014), 032822.
[21]	P. Krejčí and Ph. Laurençot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183.
[22]	P. Krejčí, J. P. O'Kane, A. Pokrovskii and D. Rachinskii}, Properties of solutions to a class of differential models incorporating Preisach hysteresis operator, Phys. D, 241 (2012), 2010-2028. doi: 10.1016/j.physd.2011.05.005.
[23]	P. Krejčí, J. P. O'Kane, A. Pokrovskii and D. Rachinskii, Stability results for a soil model with singular hysteretic hydrology, J. Phys.: Conf. Ser. 268 (2011), 012016.
[24]	K. Kuhnen, Modeling, identification and compensation of complex hysteretic nonlinearities: A modified Prandtl-Ishlinskii approach, Eur. J. Control, 9 (2003), 407-418. doi: 10.3166/ejc.9.407-418.
[25]	J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J., 7 (1957), 418-449.
[26]	Lemaitre and J.-L. Chaboche, Mechanics of Solid Materials, Cambridge University Press, Cambridge, 1990.
[27]	I. Mayergoyz, Mathematical Models of Hysteresis and Their Applications, Elsevier, 2003.
[28]	L. Prandtl, Ein Gedankenmodell zur kinetischen Theorie der festen Körper, (German) [A model for kinetic theory of solid bodies], ZAMM Z. Ang. Math. Mech., 8 (1928), 85-106.
[29]	Š. Schwabik, On a modified sum integral of Stieltjes type, Časopis Pěst. Mat., 98 (1973), 274-277.
[30]	M. Tvrdý, Regulated functions and the Perron-Stieltjes integral, Časopis Pêst. Mat., 114 (1989), 187-209.
[31]	A. Visintin, Differential Models of Hysteresis, Springer, 1994. doi: 10.1007/978-3-662-11557-2.

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