Regulation of births for viability of populations governed by age-structured problems

We assume that the evolution of the population is governed by a controlled McKendrick age-structured partial differential equation where the mortality rate and immigrated population levels are no longer known coefficients, but contingent regulation parameters chosen for a given purpose, for instance, for requiring that the population satisfies prescribed viability constraints depending on time and age. The Lotka renewal equation relating the boundary condition (number of births) to the integral with respect to age of the population is replaced by the introduction of another regulation parameter in the boundary condition, regarded as a natality policy. We may control it by its derivative, regarded as a natality decision. We then construct a regulation map, associating with the population level, the time and the age the subset of natality policies, a mortality rates and an immigration levels needed for governing viable evolutions of the population.