Optimization by Pontryagin's maximum principle on the analog computer



A computational method has been developed for obtaining the solution to a class of optimization problems by the combined use of the maximum principle and a maximum (or minimum) seeking technique on the analogue computer. Various maximum seeking techniques can be used for this method. However if the random search technique is used, this computational method has the advantage of being able to investigate a large number of operating (or controlling) variables.

The calculation procedure is essentially a trial-and-error procedure which is alternately integration and maximum seeking operations. The variables over which the system is to be optimized are approximated by a finite number of straight line segments n. Thus the maximum (or minimum) of the Hamiltonian function which is obtained by the maximum principle need only be obtained at n + 1 points along the optimization path (or trajectory).

To illustrate the use of the method the optimum operating variable profiles (or gradients) in a tubular chemical reactor were computed.

The present method, in addition to making it easier to investigate systems with a moderate number of state variables, can be used to solve problems with almost any kind of constraints and performance index encountered in ordinary optimum design problems. It offers some possibilities for on-line optimizing control of a process. A special purpose analogue computer could be built for this use.