Discrete verification of necessary conditions for switched nonlinear optimal control systems

We consider a fairly general class of state-constrained nonlinear hybrid optimal control problems that are based on coordinatizing Sussmann´s model. An event set generalizes the notion of a guard set, reset map, endpoint set as well as the switching set. We present a pseudospectral (PS) knotting method that discretizes the continuous-time variables of the problem. The discrete event conditions are imposed over the PS knots leading to a large, sparse, mixed-variable programming (MVP) problem. The Karush-Kuhn-Tucker conditions for the MVP are transformed in a manner that makes them closely resemble the discretized necessary conditions obtained from the hybrid minimum principle. A set of closure conditions are introduced to facilitate commuting the operations of dualization and discretization. An immediate consequence of this is a hybrid covector mapping theorem that provides an order-preserving transformation of the Lagrange multipliers associated with the discretized problem to the discretized covectors associated with the hybrid optimal control problem.