Article
MSC:
34C10,
34C11,
34C55,
34D40,
47H30,
82D40 | MR 1800964 | Zbl 1010.34038 | DOI: 10.1023/A:1022333500777
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Keywords:
Preisach model; hysteresis; forced oscillations; asymptotic behavior
Preisach model; hysteresis; forced oscillations; asymptotic behavior
Summary:
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 $.
References:
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