Outwards pointing hysteresis operators and asymptotic behaviour of evolution equations.
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.
scientific article; zbMATH DE number 2001953