Piecewise affine systems controllability and hybrid optimal control

We consider a particular class of hybrid systems, defined by a piecewise affine dynamic over non-overlapping regions of the state space. We want to control their behaviors so that it reaches a target by minimizing a given cost. We provide a new numerical algorithm under-approximating the controllable domain under the given hybrid dynamic. Given an optimal sequence of states of the hybrid automaton, we are then able to traverse the automaton till the target, locally insuring optimality.