A preconditioning theory for the standard optimal control problem

Abstract

This paper develops a preconditioning theory for the standard optimal control problem. Preconditioning is the execution of a coordinate transformation in order to increase the convergence rate of gradient based numerical algorithms. The transforming operator employed for this purpose is the square root of a symmetric positive definite matrix which is called the preconditioner. It was shown in [l] that the optimal preconditioner for nonlinear constrained mathematical programming problems is a matrix, denoted as 'H+, which is the positive definite reflection of the Hessian matrix 'H. By positive definite reflection, it is meant that matrix whose eigenvectors are identical to those of the Hessian and whose eigenvalues are the moduli of the latter's eigenvalues. It was demonstrated that the preconditioner 'H+ produces a transformed Hessian with a condition number of unity; it was further shown how H+ can be efficiently constructed from a singular value decomposition of 7-t. Since the Hessian is a function of the optimizing variables in a nonlinear problem, periodic application of the preconditioner is required for best results. Here, this preconditioning theory is extended to the standard optimal control problem using the direct approach of transcription. The optimal preconditioner, as well as suboptimal alternatives, are presented based on the Hessian of the resulting finite dimensional problem. Additionally, by allowing the discretization level to approach zero, the infinite-dimensional Hessian of the optimal control problem is obtained, which turns out to be a sparse matrix-valued hybrid surface consisting of ridges and discrete points. This is a nice geometrical result which provides great insight into the optimal control problem. It is shown for completeness, that as At -+ 0, the necessary conditions and the second differential of the mathematical programming problem converge to the necessary conditions and the second variation of the control problem. It is further shown that the second variation of the control problem has the form ;AZ~ 'HAZ, where the Hessian is a hybrid surface and Ar is an infinite-dimensional vector. For the second variation to be positive, the Hessian sur'Associate Fellow, AIAA 'Member, AIAA This paper is declared a work of the U.S. Government and is not subject to copyright protection in the United States. face must be positive definite with respect to surface multiplication, surface multiplication being the infinite-dimensional extension of matrix multiplication. Finally, it is noted that the extended theory is also valid in the indirect approach to optimal control provided a new integration implementation technique is employed, as shown in the separate work [2].