A maximum principle for hybrid optimal control problems

Presents a version of the maximum principle for hybrid optimal control problems under weak regularity conditions. In particular, we only consider autonomous systems, in which the dynamical behavior and the cost are invariant under time translations. The maximum principle is stated as a general assertion involving terms that are not yet precisely defined, and without a detailed specification of technical assumptions. One version of the principle, where the terms are precisely defined and the appropriate technical requirements are completely specified, is stated for problems where all the basic objects-the dynamics, the Lagrangian and the cost functions for the switchings and the end-point constraints-are differentiable along the reference arc. Another version, involving nonsmooth maps, is also stated, and some brief remarks on even more general versions are given. To illustrate the use of the maximum principle, two very simple examples are shown, involving problems that can easily be solved directly. Our results are stronger than the usual versions of the finite-dimensional maximum principle. For example, even the theorem for classical differentials applies to situations where the maps are not of class C¹, and can fail to be Lipschitz-continuous. The nonsmooth result applies to maps that are neither Lipschitz-continuous nor differentiable in the classical sense. In each case, it would be trivial to construct hybrid examples of a similar nature. On the other hand, the results presented in this paper are much weaker than what can actually be proved by our methods