Relaxation methods for nonlinear dynamics and hysteresis operators

摘要

Previous research has demonstrated that rigorous modeling and identification theory can be derived for structural dynamical models that incorporate control influence operators that are static Krasnoselskii-Pokrovskii integral hysteresis operators. Experimental evidence likewise has shown that some dynamic hysteresis models provide more accurate representations of a class of structural systms actuated by some active materials including shape memory alloys and piezoceramics. In this paper, we show that the representation of control influence operators via static hysteresis operators can be interpreted in terms of a homogeneous Young's measure. Within this framework, we subsequently derive dynamic hysteresis operators represented in terms of Young's measures that are parameterized in time. We show that the resulting integrodifferential equations are similar to the class of relaxed controls discussed by Warga [10], Gamkrelidze [24], and Roubicek [25].