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 & Continuous Dynamical Systems - B, 2017, 22 (10) : 3783-3795. doi: 10.3934/dcdsb.2017190
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[17]

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

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[17]

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

[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.  Google Scholar

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)
[1]

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

[2]

Dmitrii Rachinskii. On geometric conditions for reduction of the Moreau sweeping process to the Prandtl-Ishlinskii operator. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3361-3386. doi: 10.3934/dcdsb.2018246

[3]

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

[4]

Mario Ahues, Filomena D. d'Almeida, Alain Largillier, Paulo B. Vasconcelos. Defect correction for spectral computations for a singular integral operator. Communications on Pure & Applied Analysis, 2006, 5 (2) : 241-250. doi: 10.3934/cpaa.2006.5.241

[5]

Sanyi Tang, Wenhong Pang. On the continuity of the function describing the times of meeting impulsive set and its application. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1399-1406. doi: 10.3934/mbe.2017072

[6]

Y. T. Li, R. Wong. Integral and series representations of the dirac delta function. Communications on Pure & Applied Analysis, 2008, 7 (2) : 229-247. doi: 10.3934/cpaa.2008.7.229

[7]

Peter Giesl. Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 101-124. doi: 10.3934/dcdsb.2007.7.101

[8]

Carlos Conca, Luis Friz, Jaime H. Ortega. Direct integral decomposition for periodic function spaces and application to Bloch waves. Networks & Heterogeneous Media, 2008, 3 (3) : 555-566. doi: 10.3934/nhm.2008.3.555

[9]

Radjesvarane Alexandre, Lingbing He. Integral estimates for a linear singular operator linked with Boltzmann operators part II: High singularities $1\le\nu<2$. Kinetic & Related Models, 2008, 1 (4) : 491-513. doi: 10.3934/krm.2008.1.491

[10]

Lianwang Deng. Local integral manifolds for nonautonomous and ill-posed equations with sectorially dichotomous operator. Communications on Pure & Applied Analysis, 2020, 19 (1) : 145-174. doi: 10.3934/cpaa.2020009

[11]

Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247

[12]

E. Trofimchuk, Sergei Trofimchuk. Global stability in a regulated logistic growth model. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 461-468. doi: 10.3934/dcdsb.2005.5.461

[13]

Ethan Akin. On chain continuity. Discrete & Continuous Dynamical Systems - A, 1996, 2 (1) : 111-120. doi: 10.3934/dcds.1996.2.111

[14]

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

[15]

Vincenzo Recupero. Hysteresis operators in metric spaces. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 773-792. doi: 10.3934/dcdss.2015.8.773

[16]

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

[17]

Xianhua Tang, Xingfu Zou. A 3/2 stability result for a regulated logistic growth model. Discrete & Continuous Dynamical Systems - B, 2002, 2 (2) : 265-278. doi: 10.3934/dcdsb.2002.2.265

[18]

Tomas Alarcon, Philipp Getto, Anna Marciniak-Czochra, Maria dM Vivanco. A model for stem cell population dynamics with regulated maturation delay. Conference Publications, 2011, 2011 (Special) : 32-43. doi: 10.3934/proc.2011.2011.32

[19]

Jana Kopfová. Nonlinear semigroup methods in problems with hysteresis. Conference Publications, 2007, 2007 (Special) : 580-589. doi: 10.3934/proc.2007.2007.580

[20]

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

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (14)
  • HTML views (3)
  • Cited by (0)

Other articles
by authors

[Back to Top]