Exactly solving the weighted time/fuel optimal control of an undamped harmonic oscillator

The exact solution to the problem of the weighted time/fuel optimal control of an undamped harmonic oscillator with one bounded control and any initial state is summarized. The motivation is the possibility of improvements in the final behavior of trajectories produced by Vander Velde's designs for on/off controls of large space structures, by replacing his approximate solution with the exact one. This includes: 1) the development and extensive study of the first-order necessary conditions; 2) investigations of the existence, normality, uniqueness, and determination of the extremals and the optimal solution; and 3) numerical comparisons between the approximate and exact solutions according to three criteria, including those improvements. It is found that the extremals (solution) exist(s), are (is) normal, can be nonunique but in a finite number (is unique), can be determined by an algorithm based on items 1 and 2, and seems to improve the final behavior of the controller.