Hybrid parametric minimum principle

This paper discusses a special kind of hybrid system, named hybrid parametric system, which subsystem is characterized by parameters of model function. The parameters of model function represent the batch operation (discrete control). The joint optimization of both continuous and discrete control variables is named hybrid parametric optimal control problem. The corresponding necessary optimality conditions are established in the form of the hybrid parametric minimum principles, which are proposed and proved in this paper. The first principle, i.e. single stage hybrid parametric minimum principle, is proved based on the similar technique used by the proof of the classical Pontryagin maximum principle, and the second principle, multistage hybrid parametric minimum principle, is proved by reducing it into a nominal case of first principle. Moreover, in the virtue of the continuity of parameter space, the hybrid parametric minimum principle only requires a prescribed number of switching times.