Using Pontryagin's Maximum Principle to Solve One-Dimensional Optimization Problems, with and without Constraints, on an Iterative Analog Computer

Abstract

An analog computer technique is presented which enables application of Pontryagin's maximum prin ciple to the problem of optimizing control systems. The key problem in using Pontryagin's maximum principle is the extremization of the Hamiltonian function at every instant of time. Since the analog computer is an excellent differential equation solver, it is of advantage to convert this task into a dynamic problem. The technique used to do this is based upon the steepest ascent method. The method is applied to a one-dimensional control problem; higher-di mensional control problems can be treated using the same approach. The argument that an analog computer can solve differential equations with only one independent variable, corresponding to machine time, is true only in a technical sense. In practice it is feasible for cer tain types of problems to integrate one set of differ ential equations sufficiently fast enough so that, while integrating another set of differential equations at a much slower rate, the solution error associated with this approach remains within acceptable limits. When using the analog computer in this way, one time domain always corresponds to the solution time required for solving the differential equations de scribing the system; a second time domain corre sponds to the solution time required for solving an auxiliary set of differential equations which has no direct relationship with the system. Technological improvements and innovations made in the analog computer field during the recent past have contributed to the successful application of this approach.