Unification of the direct and indirect approaches to optimal control

This paper analyzes the direct and indirect approaches to optimal control via an ordinary gradientrestoration algorithm. First, using the indirect approach, a non-implementable infinite dimensional version of the algorithm, designed to converge to the infinite dimensional necessary conditions of the control problem, is finite-dimensionalized using generic high-order integration schemes. Second, using the direct approach, a finite dimensional version of the algorithm, designed to solve mathematical progranming problems, is applied to the mathematical programming problem which results from the direct transcription of the control problem via the same integration method. By comparing the two algorithms, it is seen how the application of the integration methods to the control problem versus the application of the integration methods to the control problem’s necessary conditions ultimately results in the solution of two different mathematical programming problems. Using an auxillary optimal control problem it is then shown how to implement the infinite dimensional a,gorithm developed using the indirect approach, so that the two finite dimensional algorithms are solving the same problem. This is an important result which unifies the direct and the indirect approaches. It has further significance in that it is now possible, using the indirect approach, to know the exact mathematical programming problem which is being solved, which allows one to accurately compute the Hessian. Exact knowledge of the Hessian is important in preconditioning ill-conditioned problems to enhance the convergence rate of gradient based dgorithms.