An Analysis of Discretization in the Indirect Approach to Optimal Control Algorithmic Development

In the indirect approach to developing algorithms for the numerical solution of optimal control problems [1]-[8], the infinite-dimensionality of the problem is retained. This is accomplished by developing a non-implementable algorithm which involves functions, integrals, and differential equations which is designed to iteratively converge to the solution of the infinite-dimensional necessary conditions. It is only after theoretic development and for implementation purposes that the resulting algorithm is then discretized sad high-order integration methods such as Runge-Kutta and quadrature are incorporated. At this point, the algorithm has been reduced to a finite-dimensional one and what is actually being solved is a mathematical programming problem. In this paper, an ordinary gradient-restoration algorithm is utilized to show that, except for the case of simple Euler integration, the resulting mathematical programming problem is extremely complicated and for all practical purposes non-identifiable. This is due to the fact that the sophisticated integration schemes are being applied directly to the differential adjoint equations as well as to the linearized constraint equations. Thus, even though algorithms developed using the indirect approach have been highly successful in providing approximate solutions to numerous optimal control problems, there is no way of determining the exact finite-dimensional problem which is being solved. Lack of such information negatively impacts applications to control problems which are ill-conditioned, since it is impossible to exactly determine the finite-dimensional Hessian. Thus, corrective preconditioning or generalized scaling [9] can only be carried out in an ad hoc suboptimal fashion.