American Institute of Mathematical Sciences

Advanced
July  2011, 29(3): 1041-1083. doi: 10.3934/dcds.2011.29.1041

Periodic solutions of parabolic problems with hysteresis on the boundary

Pavel Gurevich 1,

1. 

Free University Berlin - Institute for Mathematics 1, Arnimallee 2-6, 14195 Berlin, Germany

Received  January 2010 Revised  June 2010 Published  November 2010

We consider a parabolic problem with discontinuous hysteresis on the boundary, arising in modelling various thermal control processes. By reducing the problem to an infinite dynamical system, sufficient conditions for the existence and uniqueness of a periodic solution are found. Global stability of the periodic solution is proved.
Keywords: Hysteresis, Parabolic problems, Global attractor., Periodic solutions.
Mathematics Subject Classification: Primary: 35K10, 47J40, 35B10; Secondary: 35B4.
Citation: Pavel Gurevich. Periodic solutions of parabolic problems with hysteresis on the boundary. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1041-1083. doi: 10.3934/dcds.2011.29.1041
References:
[1]

M. S. Agranovich, On series in root vectors of operators defined by forms with a selfadjoint principal part,, Funktsional. Anal. i Prilozhen., 28 (1994), 1.   Google Scholar

[2]

H. W. Alt, On the thermostat problem,, Control Cyb., 14 (1985), 171.   Google Scholar

[3]

P.-A. Bliman and A. M. Krasnosel'skii, Periodic solutions of linear systems coupled with relay,, in, 30 (1997), 687.  doi: 10.1016/S0362-546X(96)00372-0.  Google Scholar

[4]

M. Brokate and A. Friedman, Optimal design for heat conduction problems with hysteresis,, SIAM J. Control Opt., 27 (1989), 697.  doi: 10.1137/0327037.  Google Scholar

[5]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Springer, (1996).   Google Scholar

[6]

P. Colli, M. Grasselli and J. Sprekels, Automatic control via thermostats of a hyperbolic Stefan problem with memory,, Appl. Math. Optim., 39 (1999), 229.  doi: 10.1007/s002459900105.  Google Scholar

[7]

M. Fečkan, Periodic solutions in systems at resonances with small relay hysteresis,, Math. Slovaca, 49 (1999), 41.   Google Scholar

[8]

A. Friedman and K.-H. Hoffmann, Control of free boundary problems with hysteresis,, SIAM J. Control. Optim., 26 (1988), 42.  doi: 10.1137/0326003.  Google Scholar

[9]

A. Friedman and L.-S. Jiang, Periodic solutions for a thermostat control problem,, Commun. Partial Differential Equations, 13 (1988), 515.  doi: 10.1080/03605308808820551.  Google Scholar

[10]

K. Glashoff and J. Sprekels, An application of Glicksberg's theorem to set-valued integral equations arising in the theory of thermostats,, SIAM J. Math. Anal., 12 (1981), 477.  doi: 10.1137/0512041.  Google Scholar

[11]

K. Glashoff and J. Sprekels, The regulation of temperature by thermostats and set-valued integral equations,, J. Integral Equ., 4 (1982), 95.   Google Scholar

[12]

I. G. Götz, K.-H. Hoffmann and A. M. Meirmanov, Periodic solutions of the Stefan problem with hysteresis-type boundary conditions,, Manuscripta Math., 78 (1983), 179.  doi: 10.1007/BF02599308.  Google Scholar

[13]

P. L. Gurevich and W. Jäger, Parabolic problems with the Preisach hysteresis operator in boundary conditions,, J. Differential Equations, 47 (2009), 2966.  doi: 10.1016/j.jde.2009.07.033.  Google Scholar

[14]

P. L. Gurevich, W. Jäger and A. L. Skubachevskii, On periodicity of solutions for thermocontrol problems with hysteresis-type switches,, SIAM J. Math. Anal., 41 (2009), 733.  doi: 10.1137/080718905.  Google Scholar

[15]

K.-H. Hoffmann, M. Niezgódka and J. Sprekels, Feedback control via thermostats of multidimensional two-phase Stefan problems,, Nonlinear Anal., 15 (1990), 955.  doi: 10.1016/0362-546X(90)90078-U.  Google Scholar

[16]

N. Kenmochi and A. Visintin, Asymptotic stability for nonlinear PDEs with hysteresis,, European J. Appl. Math., 5 (1994), 39.   Google Scholar

[17]

M. A. Krasnosel'skii and A. V. Pokrovskii, "Systems with Hysteresis,", Springer-Verlag, (1989).   Google Scholar

[18]

P. Krejči, J. Sprekels and U. Stefanelli, Phase-field models with hysteresis in one-dimensional thermo-visco-plasticity,, SIAM J. Math. Anal., 34 (2002), 409.  doi: 10.1137/S0036141001387604.  Google Scholar

[19]

V. B. Lidskii, Summability of series in terms of the principal vectors of non-selfadjoint operators,, Trudy Moskov. Mat. Obsc., 11 (1962), 3.   Google Scholar

[20]

J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications, Vol. I,", Springer, (1972).   Google Scholar

[21]

J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications, Vol. II,", Springer, (1972).   Google Scholar

[22]

J. Macki, P. Nistri and P. Zecca, Mathematical models for hysteresis,, SIAM Rev., 35 (1993), 94.  doi: 10.1137/1035005.  Google Scholar

[23]

G. S. Osipenko, M. V. Senkov and S. B. Tikhomirov, Algorithms of construction of invariant manifolds and attractors,, Abstracts Intern. Conf., 101 (2005).   Google Scholar

[24]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Appl. Math. Sci., 44 (1983).   Google Scholar

[25]

V. V. Pod"yapol'skii, Completeness of a system of root functions of a nonlocal problem in $L_p$,, Mat. Zametki, 71 (2002), 878.  doi: 10.1023/A:1015872912925.  Google Scholar

[26]

J. Prüss, Periodic solutions of the thermostat problem,, in Proc. Conf., 1223 (1986), 216.   Google Scholar

[27]

T. I. Seidman, Switching systems and periodicity,, in Proc. Conf., 1394 (1989), 199.   Google Scholar

[28]

S. Varigonda and T. Georgiou, Dynamics of relay relaxation oscillators,, IEEE Trans. Automat. Control, 46 (2001), 65.  doi: 10.1109/9.898696.  Google Scholar

[29]

A. Visintin, "Differential Models of Hysteresis,", Springer-Verlag, (1994).   Google Scholar

[30]

A. Visintin, Quasilinear parabolic P.D.E.s with discontinuous hysteresis,, Annali di Matematica, 185 (2006), 487.  doi: 10.1007/s10231-005-0164-6.  Google Scholar

[31]

L. F. Xu, Two parabolic equations with hysteresis,, J. Partial Differential Equations, 4 (1991), 51.   Google Scholar

show all references

References:
[1]

M. S. Agranovich, On series in root vectors of operators defined by forms with a selfadjoint principal part,, Funktsional. Anal. i Prilozhen., 28 (1994), 1.   Google Scholar

[2]

H. W. Alt, On the thermostat problem,, Control Cyb., 14 (1985), 171.   Google Scholar

[3]

P.-A. Bliman and A. M. Krasnosel'skii, Periodic solutions of linear systems coupled with relay,, in, 30 (1997), 687.  doi: 10.1016/S0362-546X(96)00372-0.  Google Scholar

[4]

M. Brokate and A. Friedman, Optimal design for heat conduction problems with hysteresis,, SIAM J. Control Opt., 27 (1989), 697.  doi: 10.1137/0327037.  Google Scholar

[5]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Springer, (1996).   Google Scholar

[6]

P. Colli, M. Grasselli and J. Sprekels, Automatic control via thermostats of a hyperbolic Stefan problem with memory,, Appl. Math. Optim., 39 (1999), 229.  doi: 10.1007/s002459900105.  Google Scholar

[7]

M. Fečkan, Periodic solutions in systems at resonances with small relay hysteresis,, Math. Slovaca, 49 (1999), 41.   Google Scholar

[8]

A. Friedman and K.-H. Hoffmann, Control of free boundary problems with hysteresis,, SIAM J. Control. Optim., 26 (1988), 42.  doi: 10.1137/0326003.  Google Scholar

[9]

A. Friedman and L.-S. Jiang, Periodic solutions for a thermostat control problem,, Commun. Partial Differential Equations, 13 (1988), 515.  doi: 10.1080/03605308808820551.  Google Scholar

[10]

K. Glashoff and J. Sprekels, An application of Glicksberg's theorem to set-valued integral equations arising in the theory of thermostats,, SIAM J. Math. Anal., 12 (1981), 477.  doi: 10.1137/0512041.  Google Scholar

[11]

K. Glashoff and J. Sprekels, The regulation of temperature by thermostats and set-valued integral equations,, J. Integral Equ., 4 (1982), 95.   Google Scholar

[12]

I. G. Götz, K.-H. Hoffmann and A. M. Meirmanov, Periodic solutions of the Stefan problem with hysteresis-type boundary conditions,, Manuscripta Math., 78 (1983), 179.  doi: 10.1007/BF02599308.  Google Scholar

[13]

P. L. Gurevich and W. Jäger, Parabolic problems with the Preisach hysteresis operator in boundary conditions,, J. Differential Equations, 47 (2009), 2966.  doi: 10.1016/j.jde.2009.07.033.  Google Scholar

[14]

P. L. Gurevich, W. Jäger and A. L. Skubachevskii, On periodicity of solutions for thermocontrol problems with hysteresis-type switches,, SIAM J. Math. Anal., 41 (2009), 733.  doi: 10.1137/080718905.  Google Scholar

[15]

K.-H. Hoffmann, M. Niezgódka and J. Sprekels, Feedback control via thermostats of multidimensional two-phase Stefan problems,, Nonlinear Anal., 15 (1990), 955.  doi: 10.1016/0362-546X(90)90078-U.  Google Scholar

[16]

N. Kenmochi and A. Visintin, Asymptotic stability for nonlinear PDEs with hysteresis,, European J. Appl. Math., 5 (1994), 39.   Google Scholar

[17]

M. A. Krasnosel'skii and A. V. Pokrovskii, "Systems with Hysteresis,", Springer-Verlag, (1989).   Google Scholar

[18]

P. Krejči, J. Sprekels and U. Stefanelli, Phase-field models with hysteresis in one-dimensional thermo-visco-plasticity,, SIAM J. Math. Anal., 34 (2002), 409.  doi: 10.1137/S0036141001387604.  Google Scholar

[19]

V. B. Lidskii, Summability of series in terms of the principal vectors of non-selfadjoint operators,, Trudy Moskov. Mat. Obsc., 11 (1962), 3.   Google Scholar

[20]

J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications, Vol. I,", Springer, (1972).   Google Scholar

[21]

J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications, Vol. II,", Springer, (1972).   Google Scholar

[22]

J. Macki, P. Nistri and P. Zecca, Mathematical models for hysteresis,, SIAM Rev., 35 (1993), 94.  doi: 10.1137/1035005.  Google Scholar

[23]

G. S. Osipenko, M. V. Senkov and S. B. Tikhomirov, Algorithms of construction of invariant manifolds and attractors,, Abstracts Intern. Conf., 101 (2005).   Google Scholar

[24]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Appl. Math. Sci., 44 (1983).   Google Scholar

[25]

V. V. Pod"yapol'skii, Completeness of a system of root functions of a nonlocal problem in $L_p$,, Mat. Zametki, 71 (2002), 878.  doi: 10.1023/A:1015872912925.  Google Scholar

[26]

J. Prüss, Periodic solutions of the thermostat problem,, in Proc. Conf., 1223 (1986), 216.   Google Scholar

[27]

T. I. Seidman, Switching systems and periodicity,, in Proc. Conf., 1394 (1989), 199.   Google Scholar

[28]

S. Varigonda and T. Georgiou, Dynamics of relay relaxation oscillators,, IEEE Trans. Automat. Control, 46 (2001), 65.  doi: 10.1109/9.898696.  Google Scholar

[29]

A. Visintin, "Differential Models of Hysteresis,", Springer-Verlag, (1994).   Google Scholar

[30]

A. Visintin, Quasilinear parabolic P.D.E.s with discontinuous hysteresis,, Annali di Matematica, 185 (2006), 487.  doi: 10.1007/s10231-005-0164-6.  Google Scholar

[31]

L. F. Xu, Two parabolic equations with hysteresis,, J. Partial Differential Equations, 4 (1991), 51.   Google Scholar

[1]

Hiroshi Matano, Ken-Ichi Nakamura. The global attractor of semilinear parabolic equations on $S^1$. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 1-24. doi: 10.3934/dcds.1997.3.1
[2]

E. N. Dancer, Norimichi Hirano. Existence of stable and unstable periodic solutions for semilinear parabolic problems. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 207-216. doi: 10.3934/dcds.1997.3.207
[3]

Pavol Quittner, Philippe Souplet. A priori estimates of global solutions of superlinear parabolic problems without variational structure. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1277-1292. doi: 10.3934/dcds.2003.9.1277
[4]

Youssef Amal, Martin Campos Pinto. Global solutions for an age-dependent model of nucleation, growth and ageing with hysteresis. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 517-535. doi: 10.3934/dcdsb.2010.13.517
[5]

Jana Kopfová. Nonlinear semigroup methods in problems with hysteresis. Conference Publications, 2007, 2007 (Special) : 580-589. doi: 10.3934/proc.2007.2007.580
[6]

Ning Ju. The global attractor for the solutions to the 3D viscous primitive equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 159-179. doi: 10.3934/dcds.2007.17.159
[7]

Elisabetta Rocca, Giulio Schimperna. Global attractor for a parabolic-hyperbolic Penrose-Fife phase field system. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1193-1214. doi: 10.3934/dcds.2006.15.1193
[8]

Maria Assunta Pozio, Fabio Punzo, Alberto Tesei. Uniqueness and nonuniqueness of solutions to parabolic problems with singular coefficients. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 891-916. doi: 10.3934/dcds.2011.30.891
[9]

Maria Assunta Pozio, Alberto Tesei. On the uniqueness of bounded solutions to singular parabolic problems. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 117-137. doi: 10.3934/dcds.2005.13.117
[10]

Kazuhiro Ishige, Paolo Salani. On a new kind of convexity for solutions of parabolic problems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 851-864. doi: 10.3934/dcdss.2011.4.851
[11]

Shiqing Zhang, Qing Zhou. Nonplanar and noncollision periodic solutions for $N$-body problems. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 679-685. doi: 10.3934/dcds.2004.10.679
[12]

Samir Adly, Daniel Goeleven, Dumitru Motreanu. Periodic and homoclinic solutions for a class of unilateral problems. Discrete & Continuous Dynamical Systems - A, 1997, 3 (4) : 579-590. doi: 10.3934/dcds.1997.3.579
[13]

Emil Minchev, Mitsuharu Ôtani. $L^∞$-energy method for a parabolic system with convection and hysteresis effect. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1613-1632. doi: 10.3934/cpaa.2018077
[14]

Tetsutaro Shibata. Global behavior of bifurcation curves for the nonlinear eigenvalue problems with periodic nonlinear terms. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2139-2147. doi: 10.3934/cpaa.2018102
[15]

Changrong Zhu, Bin Long. The periodic solutions bifurcated from a homoclinic solution for parabolic differential equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3793-3808. doi: 10.3934/dcdsb.2016121
[16]

Boling Guo, Guoli Zhou. Finite dimensionality of global attractor for the solutions to 3D viscous primitive equations of large-scale moist atmosphere. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4305-4327. doi: 10.3934/dcdsb.2018160
[17]

Zhengce Zhang, Yan Li. Global existence and gradient blowup of solutions for a semilinear parabolic equation with exponential source. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 3019-3029. doi: 10.3934/dcdsb.2014.19.3019
[18]

Davide Guidetti. Classical solutions to quasilinear parabolic problems with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 717-736. doi: 10.3934/dcdss.2016024
[19]

Davide Guidetti. Convergence to a stationary state of solutions to inverse problems of parabolic type. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 711-722. doi: 10.3934/dcdss.2013.6.711
[20]

Piotr Biler, Ignacio Guerra, Grzegorz Karch. Large global-in-time solutions of the parabolic-parabolic Keller-Segel system on the plane. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2117-2126. doi: 10.3934/cpaa.2015.14.2117

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (27)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences