The weak maximum principle for hybrid systems

The aim of this paper is to formulate the weak maximum principle for hybrid optimal control problems. We give definitions of a hybrid system and a hybrid state trajectory. These definitions are kept at an equation-based level, to make the analysis helpful while constructing an appropriate numerical procedure. We then define an optimal control problem for a hybrid system. Next, we analyze the perturbations of a hybrid state trajectory in response to variations of a control function. We build first order approximations to the trajectory perturbations and utilize them to approximate the variations of cost and constraint functionals in the optimal control problem. Finally, we introduce the adjoint variables, which enables to formulate the weak maximum principle in an elegant way. We also give some remarks concerning a construction of a numerical procedure for hybrid optimal control problems.