Minimum-energy control of a second-order nonlinear system

This paper establishes the bounded control function 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 . The performance index to be minimized is given by , with open and The analysis is accomplished with the use of the Pontryagin maximum principle. It is established that singular controls can result when . is the maximum value of .