Outwards pointing hysteresis operators and asymptotic behaviour of evolution equations.
type: Item
instance of: scholarly article
title: Outwards pointing hysteresis operators and asymptotic behaviour of evolution equations.
zbMATH Open document ID: 1043.47047
DOI: 10.1016/S1468-1218(03)00013-0
author: Olaf Klein
author: Pavel Krejč{í }
published in: Nonlinear Analysis. Real World Applications
publication date: 2003-11-10T00:00:00Z
review text: The authors study the long-time behavior of systems described by ordinary and partial differential equations with hysteresis. For their purpose, they suppose that the hysteresis operator has the so-called ``outward pointing property''. Roughly speaking, this means that if, during a time interval \([0,t]\), the input \(u\) evolves inside a neighborhood of a suitable interval of values \([A,B]\), and if \(u(t)\) exceeds \([A,B]\) (from one side), then the final value \(w(t)\) of the output exceeds a suitable fixed neighborhood of the value zero (from the same side). They give sufficient conditions in order that the generalized Prandtl-Ishlinskii operator (a suitable ``perturbation'' of the classical Prandtl-Ishlinskii operator) satisfies the outward pointing property. Moreover, they also give sufficient and necessary conditions in order that the generalized Prandtl-Ishlinskii operator satisfies a suitable concept of ``thermodynamical consistency'' (which relies to the existence of an ``internal energy functional'' which, together with the operator, satisfies a suitable estimates). As an application, they study the long-time behavior of the solution of a partial differential equation describing the wave propagation in a visco-elastic-plastic one-dimensional medium. In particular, the equation contains the spatial derivative of the hysteresis component. Assuming that the hysteresis operator is outward pointing and thermodynamically consistent, they are able to prove that, under general hypotheses, there is a unique solution defined for all times. Moreover, they are also able to prove that the time derivatives of the solution (i.e., \(u_{tt}\) and \(u_{xt}\)) tend to zero in \(L^2(0,1)\) as the time goes to infinity, where \((0,1)\) is the spatial region occupied by the medium.
reviewed by: Fabio Bagagiolo
Mathematics Subject Classification ID: 47J40
Mathematics Subject Classification ID: 34C55
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 74N30
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 74K05
zbMATH DE Number: 2001953
zbMATH Keywords: hysteresis operators
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: visco-elasto-plasticity
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2001953