Optimum Control for Bounded State Variable Problem

Abstract

For a bounded state variable problem, the optimal condition is shown by Pontryagin, but it seems to be pretty troublesome to solve the actual problems through its direct application.
It is shown in this paper that the bounded state variable problem can be completely solved through the conventional maximum principle after divided into 3 sub-problems A, B and C. In problem B which is to be solved under the restricted condition that phase point runs on the boundary, the dimension of problem B decreases by one, so that we can apply the conventional maximum principle to the degenerated problem.
Each problem A, B and C is not solved independently, but some continuity condition must be satisfied at the continuity point. The continuity conditions between problem A and B and between B and C were found by the authors. Pursuit of optimal backward time trajectory must be continued until the continuity condition is not satisfied. When the pursuit process is completed, we can obtain optimal control law.
This method is not only simple but also has clear physical meaning and does not need any other principle. Two illustrative examples are explained in the last part of this paper.