A general one-dimensional search algorithm for computing approximate solutions of some non-linear optimal control problems

Barr and Gilbert (1966, 1969 b) have presented computing algorithms for converting a brood class of optimal control problems (including minimum time, and fixed-time minimum fuel, energy and effort problems) to a sequence of optimal regulator problems, using a one dimensional search of the cost variable. These Barr and Gilbert algorithms, which use quadratic programming algorithms by the same authors (1969 a) to solve the resulting optimal regulator problems, are restricted to dynamic equations linear in state by virtue of using the convexity and compactness (Neustadt 1963) and contact function (Gilbert 1966) of the reachable set

This paper extends the above approach to a class of terminal cost optimal control problems similar to those considered by Barr and Gilbert (including quite general control constraints, but only allowing initial and final state constraints), having differential equations non-linear instate and control (where the convexity-compactness results do not hold), by converting each such problem to a sequence of optimal regulator problems, with non-linear differential equations. These, in turn, are solved by one of the author's earlier algorithms (Katz 1974) that makes use of the above convexity, compactness, and contact function results by repeatedly linearizing the regulator problems. The approach of this paper differs from that of Halkin (1964 b), in that Halkin directly linearizes the original problem (e.g. converting a non-linear minimum fuel problem to a linear minimum fuel problem) and then solves the linearized version by a doubly iterative procedure

The computing algorithm presented here is based on the definition of an appropriate approximate solution of the terminal cost problem. A local-minimum convergence proof is given, which is weak in the sense that it assumes convergence of the substep algorithm (Katz 1974) for non-linear optimal regulator problems, whose convergence has not been proved. A subsequent paper (Katz and Wachtor, to appear) shows good convergence of the (overall) terminal cost problem algorithm in examples having singular arcs, with no prior knowledge of the solution or its singular nature, other than an initial upper bound on the cost.