Method for automatic costate calculation

A method for the automatic calculation of costates using only the results obtained from direct optimization techniques is presented. The approach exploits the relation between the time-varying costates and certain sensitivities of the variational cost function, a relation that also exists between the Lagrangian multipliers obtained from a direct optimization approach and the sensitivities of the associated nonlinear-programming cost function. The complete theory for treating free, control-constrained, interior-point-constrained, and state-constrained optimal control problems is presented. As a numerical example, a state constrained version of the brachistochrone problem is solved and the results are compared to the optimal solution obtained from Pontryagin's minimum principle. The agreement is found to be excellent.