Periodic solutions of parabolic problems with hysteresis on the boundary

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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

Abstract
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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 and Continuous Dynamical Systems, 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), 1-21; English transl. in Funct. Anal. Appl., 28 (1994), 151-167.
[2]	H. W. Alt, On the thermostat problem, Control Cyb., 14 (1985), 171-193.
[3]	P.-A. Bliman and A. M. Krasnosel'skii, Periodic solutions of linear systems coupled with relay, in "Proceedings of the Second World Congress of Nonlinear Analysts, Part 2 (Athens, 1996)," Nonlinear Anal., 30 (1997), 687-696. doi: 10.1016/S0362-546X(96)00372-0.
[4]	M. Brokate and A. Friedman, Optimal design for heat conduction problems with hysteresis, SIAM J. Control Opt., 27 (1989), 697-717. doi: 10.1137/0327037.
[5]	M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Springer, Berlin, 1996.
[6]	P. Colli, M. Grasselli and J. Sprekels, Automatic control via thermostats of a hyperbolic Stefan problem with memory, Appl. Math. Optim., 39 (1999), 229-255. doi: 10.1007/s002459900105.
[7]	M. Fečkan, Periodic solutions in systems at resonances with small relay hysteresis, Math. Slovaca, 49 (1999), 41-52.
[8]	A. Friedman and K.-H. Hoffmann, Control of free boundary problems with hysteresis, SIAM J. Control. Optim., 26 (1988), 42-55. doi: 10.1137/0326003.
[9]	A. Friedman and L.-S. Jiang, Periodic solutions for a thermostat control problem, Commun. Partial Differential Equations, 13 (1988), 515-550. doi: 10.1080/03605308808820551.
[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), 477-486. doi: 10.1137/0512041.
[11]	K. Glashoff and J. Sprekels, The regulation of temperature by thermostats and set-valued integral equations, J. Integral Equ., 4 (1982), 95-112.
[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), 179-199. doi: 10.1007/BF02599308.
[13]	P. L. Gurevich and W. Jäger, Parabolic problems with the Preisach hysteresis operator in boundary conditions, J. Differential Equations, 47 (2009), 2966-3010. doi: 10.1016/j.jde.2009.07.033.
[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), 733-752. doi: 10.1137/080718905.
[15]	K.-H. Hoffmann, M. Niezgódka and J. Sprekels, Feedback control via thermostats of multidimensional two-phase Stefan problems, Nonlinear Anal., 15 (1990), 955-976. doi: 10.1016/0362-546X(90)90078-U.
[16]	N. Kenmochi and A. Visintin, Asymptotic stability for nonlinear PDEs with hysteresis, European J. Appl. Math., 5 (1994), 39-56.
[17]	M. A. Krasnosel'skii and A. V. Pokrovskii, "Systems with Hysteresis," Springer-Verlag, Berlin-Heidelberg-New York, 1989; (Translated from Russian: "Sistemy s Gisterezisom," Nauka, Moscow, 1983).
[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), 409-434. doi: 10.1137/S0036141001387604.
[19]	V. B. Lidskii, Summability of series in terms of the principal vectors of non-selfadjoint operators, Trudy Moskov. Mat. Obsc., 11 (1962), 3-35.
[20]	J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications, Vol. I," Springer, Berlin-Heidelberg-New York, 1972.
[21]	J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications, Vol. II," Springer, Berlin-Heidelberg-New York, 1972.
[22]	J. Macki, P. Nistri and P. Zecca, Mathematical models for hysteresis, SIAM Rev., 35 (1993), 94-123. doi: 10.1137/1035005.
[23]	G. S. Osipenko, M. V. Senkov and S. B. Tikhomirov, Algorithms of construction of invariant manifolds and attractors, Abstracts Intern. Conf. "Fundamental Research in Technical Universities," 101 (2005).
[24]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Appl. Math. Sci., 44, Springer, New York, 1983.
[25]	V. V. Pod"yapol'skii, Completeness of a system of root functions of a nonlocal problem in $L_p$, Mat. Zametki, 71 (2002), 878-889; English transl. in Math. Notes, 71 (2002), 804-814. doi: 10.1023/A:1015872912925.
[26]	J. Prüss, Periodic solutions of the thermostat problem, in Proc. Conf. "Differential Equations in Banach Spaces" (Bologna, July 1985), Lecture Notes Math., 1223, Springer-Verlag, Berlin - New York, (1986), 216-226.
[27]	T. I. Seidman, Switching systems and periodicity, in Proc. Conf. "Nonlinear Semigroups, Partial Differential Equations and Attractors" (Washington, DC, 1987), Lecture Notes Math., 1394, Springer-Verlag, Berlin - New York, (1989), 199-210.
[28]	S. Varigonda and T. Georgiou, Dynamics of relay relaxation oscillators, IEEE Trans. Automat. Control, 46 (2001), 65-77. doi: 10.1109/9.898696.
[29]	A. Visintin, "Differential Models of Hysteresis," Springer-Verlag, Berlin - Heidelberg, 1994.
[30]	A. Visintin, Quasilinear parabolic P.D.E.s with discontinuous hysteresis, Annali di Matematica, 185 (2006), 487-519. doi: 10.1007/s10231-005-0164-6.
[31]	L. F. Xu, Two parabolic equations with hysteresis, J. Partial Differential Equations, 4 (1991), 51-65.

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), 1-21; English transl. in Funct. Anal. Appl., 28 (1994), 151-167.
[2]	H. W. Alt, On the thermostat problem, Control Cyb., 14 (1985), 171-193.
[3]	P.-A. Bliman and A. M. Krasnosel'skii, Periodic solutions of linear systems coupled with relay, in "Proceedings of the Second World Congress of Nonlinear Analysts, Part 2 (Athens, 1996)," Nonlinear Anal., 30 (1997), 687-696. doi: 10.1016/S0362-546X(96)00372-0.
[4]	M. Brokate and A. Friedman, Optimal design for heat conduction problems with hysteresis, SIAM J. Control Opt., 27 (1989), 697-717. doi: 10.1137/0327037.
[5]	M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Springer, Berlin, 1996.
[6]	P. Colli, M. Grasselli and J. Sprekels, Automatic control via thermostats of a hyperbolic Stefan problem with memory, Appl. Math. Optim., 39 (1999), 229-255. doi: 10.1007/s002459900105.
[7]	M. Fečkan, Periodic solutions in systems at resonances with small relay hysteresis, Math. Slovaca, 49 (1999), 41-52.
[8]	A. Friedman and K.-H. Hoffmann, Control of free boundary problems with hysteresis, SIAM J. Control. Optim., 26 (1988), 42-55. doi: 10.1137/0326003.
[9]	A. Friedman and L.-S. Jiang, Periodic solutions for a thermostat control problem, Commun. Partial Differential Equations, 13 (1988), 515-550. doi: 10.1080/03605308808820551.
[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), 477-486. doi: 10.1137/0512041.
[11]	K. Glashoff and J. Sprekels, The regulation of temperature by thermostats and set-valued integral equations, J. Integral Equ., 4 (1982), 95-112.
[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), 179-199. doi: 10.1007/BF02599308.
[13]	P. L. Gurevich and W. Jäger, Parabolic problems with the Preisach hysteresis operator in boundary conditions, J. Differential Equations, 47 (2009), 2966-3010. doi: 10.1016/j.jde.2009.07.033.
[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), 733-752. doi: 10.1137/080718905.
[15]	K.-H. Hoffmann, M. Niezgódka and J. Sprekels, Feedback control via thermostats of multidimensional two-phase Stefan problems, Nonlinear Anal., 15 (1990), 955-976. doi: 10.1016/0362-546X(90)90078-U.
[16]	N. Kenmochi and A. Visintin, Asymptotic stability for nonlinear PDEs with hysteresis, European J. Appl. Math., 5 (1994), 39-56.
[17]	M. A. Krasnosel'skii and A. V. Pokrovskii, "Systems with Hysteresis," Springer-Verlag, Berlin-Heidelberg-New York, 1989; (Translated from Russian: "Sistemy s Gisterezisom," Nauka, Moscow, 1983).
[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), 409-434. doi: 10.1137/S0036141001387604.
[19]	V. B. Lidskii, Summability of series in terms of the principal vectors of non-selfadjoint operators, Trudy Moskov. Mat. Obsc., 11 (1962), 3-35.
[20]	J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications, Vol. I," Springer, Berlin-Heidelberg-New York, 1972.
[21]	J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications, Vol. II," Springer, Berlin-Heidelberg-New York, 1972.
[22]	J. Macki, P. Nistri and P. Zecca, Mathematical models for hysteresis, SIAM Rev., 35 (1993), 94-123. doi: 10.1137/1035005.
[23]	G. S. Osipenko, M. V. Senkov and S. B. Tikhomirov, Algorithms of construction of invariant manifolds and attractors, Abstracts Intern. Conf. "Fundamental Research in Technical Universities," 101 (2005).
[24]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Appl. Math. Sci., 44, Springer, New York, 1983.
[25]	V. V. Pod"yapol'skii, Completeness of a system of root functions of a nonlocal problem in $L_p$, Mat. Zametki, 71 (2002), 878-889; English transl. in Math. Notes, 71 (2002), 804-814. doi: 10.1023/A:1015872912925.
[26]	J. Prüss, Periodic solutions of the thermostat problem, in Proc. Conf. "Differential Equations in Banach Spaces" (Bologna, July 1985), Lecture Notes Math., 1223, Springer-Verlag, Berlin - New York, (1986), 216-226.
[27]	T. I. Seidman, Switching systems and periodicity, in Proc. Conf. "Nonlinear Semigroups, Partial Differential Equations and Attractors" (Washington, DC, 1987), Lecture Notes Math., 1394, Springer-Verlag, Berlin - New York, (1989), 199-210.
[28]	S. Varigonda and T. Georgiou, Dynamics of relay relaxation oscillators, IEEE Trans. Automat. Control, 46 (2001), 65-77. doi: 10.1109/9.898696.
[29]	A. Visintin, "Differential Models of Hysteresis," Springer-Verlag, Berlin - Heidelberg, 1994.
[30]	A. Visintin, Quasilinear parabolic P.D.E.s with discontinuous hysteresis, Annali di Matematica, 185 (2006), 487-519. doi: 10.1007/s10231-005-0164-6.
[31]	L. F. Xu, Two parabolic equations with hysteresis, J. Partial Differential Equations, 4 (1991), 51-65.

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