December  2017, 22(10): 3783-3795. doi: 10.3934/dcdsb.2017190

Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions

1. 

College of Science, Hohai University, Nanjing 210098, China

2. 

Institute of Mathematics, Czech Academy of Sciences, Žitná 25, CZ-11567 Praha 1, Czech Republic

3. 

College of Science, Hohai University, Nanjing 210098, China

* Corresponding author: Wei Liu

The hospitality of the Hohai University in Nanjing during the second author's stay in OctoberNovember 2016 is gratefully acknowledged

Received  November 2016 Revised  April 2017 Published  July 2017

Fund Project: This work was supported by the Program of High-end Foreign Experts of the SAFEA (No. GDW20163200216). The work of the second author was partially supported by the GACR GrantČ 15-12227S and RVO: 67985840.

It is well known that the Prandtl-Ishlinskii hysteresis operator is locally Lipschitz continuous in the space of continuous functions provided its primary response curve is convex or concave. This property can easily be extended to any absolutely continuous primary response curve with derivative of locally bounded variation. Under the same condition, the Prandtl-Ishlinskii operator in the Kurzweil integral setting is locally Lipschitz continuous also in the space of regulated functions. This paper shows that the Prandtl-Ishlinskii operator is still continuous if the primary response curve is only monotone and continuous, and that it may not even be locally Hölder continuous for continuously differentiable primary response curves.

Citation: Wei Liu, Pavel Krejčí, Guoju Ye. Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3783-3795. doi: 10.3934/dcdsb.2017190
References:
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M. Al JanaidehS. Rakheja and C.-Y. Su, An analytical generalized Prandtl-Ishlinskii model inversion for hysteresis compensation in micropositioning control, IEEE/ASME Transactions on Mechatronics, 16 (2011), 734-744.  doi: 10.1109/TMECH.2010.2052366.

[2]

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R. CrossM. Grinfeld and H. Lamba, A mean-field model of investor behaviour, Journal of Physics: Conference Series, 55 (2006), 55-62.  doi: 10.1088/1742-6596/55/1/005.

[4]

R. CrossM. Grinfeld and H. Lamba, Hysteresis and Economics: Taking the economic past into account, IEEE Control Systems Magazine, 29 (2009), 30-43.  doi: 10.1109/MCS.2008.930445.

[5]

R. CrossH. McNamaraA. Pokrovskii and D. Rachinskii, A new paradigm for modelling hysteresis in macroeconomic flows, Physica B: Condensed Matter, 403 (2008), 231-236.  doi: 10.1016/j.physb.2007.08.017.

[6]

M. GrinfeldH. Lamba and R. Cross, A mesoscopic stock market model with hysteretic agents, Discrete Continuous Dynam. Systems -B, 18 (2013), 403-415.  doi: 10.3934/dcdsb.2013.18.403.

[7]

A. Yu. Ishlinskii, Some applications of statistical methods to describing deformations of bodies, Izv. AN SSSR, Techn. Ser., 9 (1944), 583-590. 

[8]

M. A. Krasnosel'skii and A. V. Pokrovskii, Systems with Hysteresis, Springer: Berlin; 1989. Russian edition: Nauka: Moscow; 1983. doi: 10.1007/978-3-642-61302-9.

[9]

P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations Gakuto Int. Series Math. Sci. & Appl. , Vol. 8, Gakkotosho, Tokyo 1996.

[10]

P. Krejčí, The Kurzweil integral and hysteresis, Journal of Physics: Conference Series, 55 (2006), 144-154.  doi: 10.1088/1742-6596/55/1/014.

[11]

P. KrejčíH. LambaS. Melnik and D. Rachinskii, Kurzweil integral representation of interacting Prandtl-Ishlinskii operators, Discrete Continuous Dynam. Systems -B, 20 (2015), 2949-2965.  doi: 10.3934/dcdsb.2015.20.2949.

[12]

P. KrejčíH. LambaG.A. Monteiro and D. Rachinskii, The Kurzweil integral in financial market modeling, Math. Bohem., 141 (2016), 261-286.  doi: 10.21136/MB.2016.18.

[13]

P. Krejčí and Ph. Laurençot, Hysteresis filtering in the space of bounded measurable functions, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 5 (2002), 755-772. 

[14]

P. Krejčí and Ph. Laurençot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183. 

[15]

K. Kuhnen, Modeling, identification and compensation of complex hysteretic nonlinearities: A modified Prandtl-Ishlinskii approach, European Journal of Control, 9 (2003), 407-418.  doi: 10.3166/ejc.9.407-418.

[16]

J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J.(7), 82 (1957), 418-449. 

[17]

L. Prandtl, Ein Gedankenmodell zur kinetischen Theorie der festen Körper, Z. Ang. Math. Mech., 8 (1928), 85-106. 

[18]

M. Sjöström and C. Visone, "Moving" Prandtl-Ishlinskii operators with compensator in a closed form, Physica B -Condensed Matter, 372 (2006), 97-100.  doi: 10.1016/j.physb.2005.10.016.

show all references

The hospitality of the Hohai University in Nanjing during the second author's stay in OctoberNovember 2016 is gratefully acknowledged

References:
[1]

M. Al JanaidehS. Rakheja and C.-Y. Su, An analytical generalized Prandtl-Ishlinskii model inversion for hysteresis compensation in micropositioning control, IEEE/ASME Transactions on Mechatronics, 16 (2011), 734-744.  doi: 10.1109/TMECH.2010.2052366.

[2]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Appl. Math. Sci. , 121, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4048-8.

[3]

R. CrossM. Grinfeld and H. Lamba, A mean-field model of investor behaviour, Journal of Physics: Conference Series, 55 (2006), 55-62.  doi: 10.1088/1742-6596/55/1/005.

[4]

R. CrossM. Grinfeld and H. Lamba, Hysteresis and Economics: Taking the economic past into account, IEEE Control Systems Magazine, 29 (2009), 30-43.  doi: 10.1109/MCS.2008.930445.

[5]

R. CrossH. McNamaraA. Pokrovskii and D. Rachinskii, A new paradigm for modelling hysteresis in macroeconomic flows, Physica B: Condensed Matter, 403 (2008), 231-236.  doi: 10.1016/j.physb.2007.08.017.

[6]

M. GrinfeldH. Lamba and R. Cross, A mesoscopic stock market model with hysteretic agents, Discrete Continuous Dynam. Systems -B, 18 (2013), 403-415.  doi: 10.3934/dcdsb.2013.18.403.

[7]

A. Yu. Ishlinskii, Some applications of statistical methods to describing deformations of bodies, Izv. AN SSSR, Techn. Ser., 9 (1944), 583-590. 

[8]

M. A. Krasnosel'skii and A. V. Pokrovskii, Systems with Hysteresis, Springer: Berlin; 1989. Russian edition: Nauka: Moscow; 1983. doi: 10.1007/978-3-642-61302-9.

[9]

P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations Gakuto Int. Series Math. Sci. & Appl. , Vol. 8, Gakkotosho, Tokyo 1996.

[10]

P. Krejčí, The Kurzweil integral and hysteresis, Journal of Physics: Conference Series, 55 (2006), 144-154.  doi: 10.1088/1742-6596/55/1/014.

[11]

P. KrejčíH. LambaS. Melnik and D. Rachinskii, Kurzweil integral representation of interacting Prandtl-Ishlinskii operators, Discrete Continuous Dynam. Systems -B, 20 (2015), 2949-2965.  doi: 10.3934/dcdsb.2015.20.2949.

[12]

P. KrejčíH. LambaG.A. Monteiro and D. Rachinskii, The Kurzweil integral in financial market modeling, Math. Bohem., 141 (2016), 261-286.  doi: 10.21136/MB.2016.18.

[13]

P. Krejčí and Ph. Laurençot, Hysteresis filtering in the space of bounded measurable functions, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 5 (2002), 755-772. 

[14]

P. Krejčí and Ph. Laurençot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183. 

[15]

K. Kuhnen, Modeling, identification and compensation of complex hysteretic nonlinearities: A modified Prandtl-Ishlinskii approach, European Journal of Control, 9 (2003), 407-418.  doi: 10.3166/ejc.9.407-418.

[16]

J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J.(7), 82 (1957), 418-449. 

[17]

L. Prandtl, Ein Gedankenmodell zur kinetischen Theorie der festen Körper, Z. Ang. Math. Mech., 8 (1928), 85-106. 

[18]

M. Sjöström and C. Visone, "Moving" Prandtl-Ishlinskii operators with compensator in a closed form, Physica B -Condensed Matter, 372 (2006), 97-100.  doi: 10.1016/j.physb.2005.10.016.

Figure 1.  The memory curves $\lambda(r)$ (the bold solid line) and $\hat \lambda(r)$ (the thin solid line)
Figure 2.  The memory curves $\lambda(r)$ (the solid line) and $\hat \lambda(r)$ (the {dashed} line)
Figure 3.  The primary response curve $\psi_1$ (the bold solid line), its derivative $\psi_1'$ (the bold dashed line), and the piecewise linear regularization $\psi_2'$ of $\psi_1'$ (the thin solid line)
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