The structure of optimum control systems and the form of the optimal control signals are discussed for minimum time, minimum fuel, and minimum energy control of linear time-invariant plants. If the control variables are bounded by 1 in magnitude, then the control signals must have the values +1 or -1 for minimum time response; +1, 0, or -1 for minimum fuel operation; +1, -1 or be linearly related to the state of the plant, between these limits, for minimum energy operation.
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M. Athanassiades (1963) studied this question.