Minimum-energy control of a second-order nonlinear system

Abstract
This paper establishes the bounded control functionu(t)which minimizes the total energy expended by a submerged vehicle (for propulsion and hotel load) in a rectilinear translation with arbitrary initial velocity, arbitrary displacement, and zero final velocity. The motion of the vehicle is determined by the nonlinear differential equation\ddot{x}+a\dot{x}|\dot{x}| = u, a > 0. The performance index to be minimized is given byS =\int_{0}^{T}(k+u\dot{x})dt, withTopen andk > 0.The analysis is accomplished with the use of the Pontryagin maximum principle. It is established that singular controls can result whenk \leq 2 \sqrt{U^{3}/a}.Uis the maximum value of|u(t)|.

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