# American Institute of Mathematical Sciences

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

## Periodic solutions of parabolic problems with hysteresis on the boundary

 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.
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

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##### 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
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