On synthesizing optimal controls

Abstract

A synthesis procedure for a wide class of optimal processes is presented. Linear processes are considered under the following types of optimization criteria and constraints: (1) Under control amplitude constraint, either minimize control effort or minimize time. (2) Under effort constraint, minimize time. (3) Under both amplitude and effort constraints, minimize time. (4) Under no constraint, minimize effort. In each case, the control system is to take the state vector from a given initial state to a given terminal state, or to a given terminal closed convex set. Control effort is understood either in the sense of maximum amplitude or of an integral of a certain function of the control. It is well known that such optimal controls can be given in terms of a solution of the adjoint system of differential equations of the original system. The main difficulty in computing the optimal control lies in finding the initial conditions of the adjoint system. By geometric arguments, a function of the initial condition is developed, and it is shown that this function attains a maximum at the desired value of the initial condition. Furthermore, the gradient of this function is shown to have a simple form, so that the maximum can be directly computed by the method of steepest ascent.