Previous |  Up |  Next


Preisach model; hysteresis; forced oscillations; asymptotic behavior
This paper deals with the asymptotic behavior as $t\rightarrow \infty $ of solutions $u$ to the forced Preisach oscillator equation $\ddot{w}(t) + u(t) = \psi (t)$, $w = u + {\mathcal P}[u]$, where $\mathcal P$ is a Preisach hysteresis operator, $\psi \in L^\infty (0,\infty )$ is a given function and $t\ge 0$ is the time variable. We establish an explicit asymptotic relation between the Preisach measure and the function $\psi $ (or, in a more physical terminology, a balance condition between the hysteresis dissipation and the external forcing) which guarantees that every solution remains bounded for all times. Examples show that this condition is qualitatively optimal. Moreover, if the Preisach measure does not identically vanish in any neighbourhood of the origin in the Preisach half-plane and $\lim _{t\rightarrow \infty } \psi (t) = 0$, then every bounded solution also asymptotically vanishes as $t\rightarrow \infty $.
[1] P.-A. Bliman: Etude Mathématique d’un Modèle de Frottement sec: Le Modèle de P. R.  Dahl. Thesis. Université de Paris IX (Paris-Dauphine), Paris and INRIA, Rocquencourt, 1990. MR 1289413
[2] P.-A.  Bliman, A. M.  Krasnosel’skiĭ, M.  Sorine and A. A.  Vladimirov: Nonlinear resonance in systems with hysteresis. Nonlinear Anal. 27 (1996), 561–577. DOI 10.1016/0362-546X(96)00032-6 | MR 1396029
[3] M. Brokate: Optimale Steuerung von gewöhnlichen Differentialgleichungen mit Nichtlinearitäten vom Hysterese-Typ. Peter Lang, Frankfurt am Main, 1987. (German) MR 1031251
[4] M. Brokate, K. Dreßler and P.  Krejčí: Rainflow counting and energy dissipation for hysteresis models in elastoplasticity. Euro. J.  Mech. A/Solids 15 (1996), 705–735. MR 1412202
[5] M.  Brokate, A. V. Pokrovskiĭ: Asymptotically stable oscillations in systems with hysteresis nonlinearities. J. Differential Equations 150 (1998), 98–123. DOI 10.1006/jdeq.1998.3492 | MR 1660262
[6] M.  Brokate, J.  Sprekels: Hysteresis and Phase Transitions. Appl. Math. Sci., Vol. 121. Springer-Verlag, New York, 1996. DOI 10.1007/978-1-4612-4048-8_5 | MR 1411908
[7] M.  Brokate, A.  Visintin: Properties of the Preisach model for hysteresis. J.  Reine Angew. Math. 402 (1989), 1–40. MR 1022792
[8] V. V. Chernorutskiĭ, D. I. Rachinskiĭ: On uniqueness of an initial-value problem for ODE with hysteresis. NoDEA 4 (1997), 391–399. DOI 10.1007/s000300050021 | MR 1458534
[9] M. A. Krasnosel’skiĭ, I. D. Mayergoyz, A. V. Pokrovskiĭ and D. I. Rachinskiĭ: Operators of hysteresis nonlinearity generated by continuous relay systems. Avtomat. i Telemekh. (1994), 49–60. (Russian) MR 1295891
[10] M. A. Krasnosel’skiĭ, A. V. Pokrovskiĭ: Systems with Hysteresis. English edition Springer 1989, Nauka, Moscow, 1983. (Russian) MR 0742931
[11] P.  Krejčí: On Maxwell equations with the Preisach hysteresis operator: the one-dimensional time-periodic case. Apl. Mat. 34 (1989), 364–374. MR 1014077
[12] P.  Krejčí: Global behaviour of solutions to the wave equation with hysteresis. Adv. Math. Sci. Appl. 2 (1993), 1–23.
[13] P.  Krejčí: Forced oscillations in Preisach systems. Physica B 275 (2000), 81–86. DOI 10.1016/S0921-4526(99)00713-9
[14] P.  Krejčí: Hysteresis, Convexity and Dissipation in Hyperbolic Equations. Gakuto Int. Ser. Math. Sci. Appl., Vol 8. Gakkōtosho, Tokyo, 1996. MR 2466538
[15] I. D.  Mayergoyz: Mathematical Models for Hysteresis. Springer-Verlag, New York, 1991. MR 1083150
[16] F.  Preisach: Über die magnetische Nachwirkung. Z. Phys. 94 (1935), 277–302. (German)
[17] A.  Visintin: On the Preisach model for hysteresis. Nonlinear Anal. 9 (1984), 977–996. DOI 10.1016/0362-546X(84)90094-4 | MR 0760191 | Zbl 0563.35007
[18] A.  Visintin: Differential Models of Hysteresis. Springer, Berlin-Heidelberg, 1994. MR 1329094 | Zbl 0820.35004
Partner of
EuDML logo