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Periodic solutions of parabolic problems with hysteresis on the boundary
1. | Free University Berlin - Institute for Mathematics 1, Arnimallee 2-6, 14195 Berlin, Germany |
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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