Relaxed Controls and a Class of Active Material Actuator Models

Abstract

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 KrasnoselskiiPokrovskii integral hysteresis operators. Experimental evidence likewise has shown that some dynamic hysteresis models provide more accurate representations of a class of structural systems actuated by some active materials including shape memory alloys and some 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 [and Roubicek [25]. The formulation presented herein differs from that studied in [10], [24] and [25] in that the kernel of the hysteresis operator is a history dependent functional, as opposed to Caratheodory integral satisfying a growth condition. The theory presented provides representations of dynamic hysteresis operators that have provided good agreement with experimental behavior in some active materials. The convergence of finite dimensional approximations of the governing equations is also discussed.