A modified maximum principle for optimum control of a system with bounded phase space coordinates

Summary In Pontryagin's maximum principle and in ‘bang-bang’ control, the manipulated variable is assumed to have no inertia, so that its position changes instantaneously from one value to another. In practice this never happens, for instance in aircraft controls the elevators and ailerons are limited in speed and displacement. The problem is a special case of the more general problem of optimal control in bounded phase space. In this paper a method is given whereby the problem is treated by a limiting process. In place of the rigid bound, a cost function with a multiplied K is introduced for regions beyond the boundaries in phase space. It is shown that in the limit of K approaching infinity, both the added cost and the maximum excursion of the optimal path beyond the boundaries approach zero, thus deriving the optimal control condition. The main results of the paper are stated in the form of two theorems. Also considered is the practical significance of the results for systems with multiple saturation.