In many applications of optimal control, the stage cost is not fixed, but
rather a design choice with considerable impact on the control performance. In
infinite horizon optimal control, the choice of stage cost is often restricted
by the requirement of uniform cost controllability, which is nontrivial to
satisfy. Here we revisit a previously proposed constructive technique for stage
cost design. We generalize its setting, weaken the required assumptions and add
additional flexibility. Furthermore, we show that the required assumptions
essentially cannot be weakened anymore. By providing improved design options
for stage costs, this work contributes to expanding the applicability of
optimization-based control methodologies, in particular, model predictive
control