  A purely state-dependent cost function can be modified by introducing a
control-dependent term rewarding submaximal control utilization. A moderation
incentive is identically zero on the boundary of the admissible control region
and non-negative on the interior; it is bounded above by the infimum of the
state-dependent cost function, so that the instantaneous total cost is always
non-negative. The conservation law determined by the Maximum Principle, in
combination with the condition that the moderation incentive equal zero on the
boundary of the admissible control region, plays a crucial role in the
analysis; in some situations, the initial and final values of the auxiliary
variable are uniquely determined by the condition that the conserved quantity
equal zero along a solution of the arbitrary duration synthesis problem. Use of
an alternate system of evolution equations, parametrized by the auxiliary
variable, for one-degree of freedom controlled acceleration systems, can
significantly simplify numerical searches for solutions of the arbitrary
duration synthesis problem. A one-parameter family of 'elliptical' moderation
incentives is introduced; the behavior of the well-known quadratic control cost
and its incentive analog is compared to that of the elliptical incentives in
two simple controlled acceleration examples. The elliptical incentives yield
smooth solutions with controls remaining in the interior of the admissible
region, while the quadratic incentive allows piecewise smooth solutions with
controls moving on and off the boundary of the admissible region; in these
examples, the arbitrary duration synthesis problem for the traditional
quadratic control cost has no solution; the total cost is a monotonically
decreasing function of the duration.
