Minimum-energy control of a second-order nonlinear system

This paper establishes the bounded control function
u(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 by
S =\int_{0}^{T}(k+u\dot{x})dt
, with
T
open and
k > 0.
The analysis is accomplished with the use of the Pontryagin maximum principle. It is established that singular controls can result when
k \leq 2 \sqrt{U^{3}/a}
.
U
is the maximum value of
|u(t)|
.