Convex Programming and Optimal Control

The use of convex programming to attack problems of optimal control is not new, but it is becoming of increasing interest. Techniques of steepest descent and gradient projection have been used by Balakrishnan [1], Goldstein [2], [3], Neustadt [4], and Neustadt and Paiewonsky [5]. For the case of unbounded fuel-optimal linear controls Neustadt [4] has shown that the problem may be cast into the form of an infinite linear program. More recently Dantzig [7] and Van Slyke [8] have obtained results in this direction for bounded linear controls. This paper will be concerned with the case of fuel-optimal linear controls. This problem will be reduced to the case of minimizing a convex function on $E_n $, and techniques of infinite convex programming will be applied. In the important case when the thrust magnitude is constrained, the convex function is continuously differentiable, and techniques of steepest descent may be applied. This approach has already been suggested by Neustadt and Paiewonsky [5].