Forced oscillations in Preisach systems

This paper gives a survey of results on the asymptotic behavior as t --> as of solutions u to the forced Preisach oscillator equation w(t) + u(t) = psi(t), w = u + P[u], where P is a Preisach hysteresis operator, psi epsilon L-infinity(0, infinity,) is a given function and t greater than or equal to 0 is the time variable. We establish an explicit asymptotic relation between the Preisach measure and the function IL (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 neighborhood of the origin in the Preisach half-plane and lim(t-->infinity) psi(t) = 0, then every bounded solution also asymptotically vanishes as t --> infinity. (C) 2000 Elsevier Science B.V. All rights reserved.