For which objective is birth process an optimal feedback in age structured population dynamics?

We consider the McKendrick linear model for the evolution of an age structured population. Usually the birth rate is given through a linear functional of the present population using the fertility rate. We are investigating the question of the existence of an objective function, depending on the control and some observation of the state, for which the associated optimal control problem using the birth rate as a control would yield the previous relation using the fertility rate as the optimal closed loop form. Then we consider adaption mechanisms that we model by including a desired value of the observation in the objective function. A modified fertility rate is derived.