A lot of papers concerned with synthesizing optimum system through the maximum principle have been published, but the detailed methods used in these papers are extremely diverse according to the various conditions such as the control time, the integral constraint, the singular solution, the initial condition and so on. The method described in this paper can be applied universally to all kinds of problems.
Synthesis means that we should constitute a feedback control system by finding the control law as the function of state variables. To begin with, the authors pointed out that almost all kinds of problems are embedded in an unfixed time problem with integral constraints. The authors clarified the synthesis procedure for such a general problem. The outline of the procedure is as follows; to describe the conditions e. g., the fixed time, limited fuel, etc., in terms of integral constraints; to specify how to pick up the state variables corresponding to the integral constraint; to examine rigorously the singular solutions that satisfy the optimum condition; to find out all the possible backtime optimum trajectories; to select the better trajectories if they intersect with one another. The completion of the procedure gives the optimum control law immediately.
The problem of finding the highest possible altitude of launched rocket with limited fuel is shown as an example in order to provide a full understanding of this method.
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