Computational Methods in Optimal Control Problems

The purpose of this modest report is to present in a simplified manner some of the computational methods that have been developed in the last ten years for the solution of optimal control problems. Only those methods that are based on the minimum (maximum) principle of Pontriagin are discussed here. The autline of the report is as follows: In the first two sections a control problem of Bolza is formulated and the necessary conditions in the form of the minimum principle are given. The method of steepest descent and a conjugate gradient-method are dis- cussed in Section 3. In the remaining sections, the successive sweep method, the Newton-Raphson method and the generalized Newton-Raphson method (also called quasilinearization method) ar presented from a unified approach which is based on the application of Newton- Raphson approximation to the necessary conditions of optimality. The second-variation method and other shooting methods based on minimizing an error function are also considered. TABLE OF CONTENTS 1. 0 INTRODUCTION 1 2. 0 NECESSARY CONDITIONS FOR OPTIMALITY -------- 2 3. 0 THE GRADIENT METHOD 4 3. 1 Min H Method and Conjugate Gradient Method -. ---------. . . . ------. --------. - 8 3. 2 Boundary Constraints -----------. ----. - 9 3. 3 Problems with Control Constraints --. -- 15 4. 0 SUCCESSIVE SWEEP METHOD -------------------- 18 4. 1 Final Time Given Implicitly ----. ------ 22 5. 0 SECOND-VARIATION METHOD -------------------- 23 6. 0 SHOOTING METHODS --------------------------- 27 6. 1 Newton-Raphson Method ----------------- 27 6.