The optimal control as a function of the instantaneous state, i.e., the optimal “feedback” or “closed-loop” control, is derived for the controlled second-order linear process with constant coefficients
for so-called minimum-fuel or minimum-effort operation (i.e., such that the time integral of the magnitude of the control u is minimized), subject to an amplitude limitation on the control |u| ⩽ L. The objective is to force the phase state from an arbitrary instantaneous value (x, x˙) to the origin within an arbitrarily prescribed time-to-run T. The solution is obtained for the nonoscillatory cases (b2 ⩾ c2 ⩾ 0) when L is finite, and for arbitrary real b and c when L is infinite; i.e., when the control is not amplitude-limited. The form of the optimal control is shown to be “bang-off-bang” with the most general initial conditions; i.e., during successive time intervals, u is constant at one limit, identically zero, and constant at the limit of opposite polarity. Explicit expressions for the switching surfaces in state space (T, x, x˙) at which u changes value and, hence, of the optimal feedback control u (T, x, x˙), are given, both with and without amplitude limitation. Without such (L = ∞) the optimal control is impulsive and the areas of the impulses in terms of the current state are obtained by a limiting procedure.
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