# Resonance in Preisach systems

• Volume: 45, Issue: 6, page 439-468
• ISSN: 0862-7940

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## Abstract

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This paper deals with the asymptotic behavior as $t\to \infty$ of solutions $u$ to the forced Preisach oscillator equation $\stackrel{¨}{w}\left(t\right)+u\left(t\right)=\psi \left(t\right)$, $w=u+𝒫\left[u\right]$, where $𝒫$ is a Preisach hysteresis operator, $\psi \in {L}^{\infty }\left(0,\infty \right)$ 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 ${lim}_{t\to \infty }\psi \left(t\right)=0$, then every bounded solution also asymptotically vanishes as $t\to \infty$.

## How to cite

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Krejčí, Pavel. "Resonance in Preisach systems." Applications of Mathematics 45.6 (2000): 439-468. <http://eudml.org/doc/33071>.

@article{Krejčí2000,
abstract = {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 $t\rightarrow \infty$.},
author = {Krejčí, Pavel},
journal = {Applications of Mathematics},
keywords = {Preisach model; hysteresis; forced oscillations; asymptotic behavior; Preisach model; hysteresis; forced oscillations; asymptotic behavior},
language = {eng},
number = {6},
pages = {439-468},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Resonance in Preisach systems},
url = {http://eudml.org/doc/33071},
volume = {45},
year = {2000},
}

TY - JOUR
AU - Krejčí, Pavel
TI - Resonance in Preisach systems
JO - Applications of Mathematics
PY - 2000
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 45
IS - 6
SP - 439
EP - 468
AB - 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 $t\rightarrow \infty$.
LA - eng
KW - Preisach model; hysteresis; forced oscillations; asymptotic behavior; Preisach model; hysteresis; forced oscillations; asymptotic behavior
UR - http://eudml.org/doc/33071
ER -

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