Legendre Pseudospectral Approximations of Optimal Control Problems

We consider nonlinear optimal control problems with mixed statecontrol constraints. A discretization of the Bolza problem by a Legendre pseudospectral method is considered. It is shown that the operations of discretization and dualization are not commutative. A set of Closure Conditions are introduced to commute these operations. An immediate consequence of this is a Covector Mapping Theorem (CMT) that provides an order-preserving transformation of the Lagrange multipliers associated with the discretized problem to the discrete covectors associated with the optimal control problem. A natural consequence of the CMT is that for pure state-constrained problems, the dual variables can be easily related to the D-form of the Lagrangian of the Hamiltonian. We demonstrate the practical advantage of our results by numerically solving a state-constrained optimal control problem without deriving the necessary conditions. The costates obtained by an application of our CMT show excellent agreement with the exact analytical solution.