We analyze a system modeling the evolution of an age and spatially structured population (of Lotka-McKendrick type). We study it by first writing it in an abstract form using several operators. We show that the semigroup associated with the corresponding system is differentiable. Using this property, we show how to prove the exponential stabilization with a finite-dimensional feedback control. We consider two types of controls: one that acts directly on the main equation of evolution and one that acts on the birth equation. One of the main difficulties in the analysis of this system is that the operators involved in the system can depend on the age variable. We use in particular a parabolic evolution operator associated with the main operator of the system. Our stabilization result shows how to extend the framework associated with parabolic system to the case of differentiable semigroups.