Geometrical Properties of Optimal Hybrid System Trajectories and the Optimization of Switching Manifolds

The geometrical properties of hybrid system optimal control (HSOC) problems are analyzed in this paper, these include (i) necessary conditions for the optimality of system trajectories based upon the geometric properties of value function gradients and of optimal controlled vector fields at switching states, (ii) value function sensitivity with respect to switching manifold displacement, and (iii) the relationship between the second fundamental forms of switching manifolds in 3 dimensional Euclidean space and the second order derivatives of HSOC value functions. Computational examples are given.