Finite difference scheme for automatic costate calculation

Abstract

A method for the automatic calculation of costates using only results obtained from direct optimization techniques is presented. The approach is based on finite differences and exploits the relation between the costates and certain sensitivities of the cost function. The complete theory for treating free, control constrained, interior-point constrained, and state constrained optimal control problems is presented. An important advantage of the method presented here is that it does not require a priori identification of the optimal switching structure. 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 Pon try a gin's minimum principle. The agreement is found to be excellent.