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This paper considers the problem arising from the design of an autopilot for a large booster. The motion-controlling actuators of the booster have both position and rate limits. The problem is formulated as a bounded phase-coordinate problem and analyzed by the ``backing out of the target'' procedure. A method of constructing the optimal control is presented. An example of an oscillatory system with two control inputs is given, and the optimal control is expressed as an explicit time function.