On the Hybrid Optimal Control Problem: Theory and Algorithms

A class of hybrid optimal control problems (HOCP) for systems with controlled and autonomous location transitions is formulated and a set of necessary conditions for hybrid system trajectory optimality is presented which together constitute generalizations of the standard Maximum Principle; these are given for the cases of open bounded control value sets and compact control value sets. The derivations in the paper employ: (i) classical variational and needle variation techniques; and (ii) a local controllability condition which is used to establish the adjoint and Hamiltonian jump conditions in the autonomous switching case. Employing the hybrid minimum principle (HMP) necessary conditions, a class of general HMP based algorithms for hybrid systems optimization are presented and analyzed for the autonomous switchings case and the controlled switchings case. Using results from the theory of penalty function methods and Ekeland's variational principle the convergence of these algorithms is established under reasonable assumptions. The efficacy of the proposed algorithms is illustrated via computational examples.