Dynamic programming for systems with hysteresis

Abstract We introduce a novel type of dynamic programming equations for controlled differential equations with hysteresis on the control. The novelty of our approach is that we use Hamilton–Jacobi functions with one of the variables being set-valued, corresponding to the active set on the Preisach plane. The derivation of these new dynamic programming equations relies on a suitable notion of derivatives of set-valued functions, which we introduce in this paper. The results have applications to systems with sensors and actuators (e.g. piezoceramic materials) that exhibit hysteresis.