We design a general set-up for the study of a generic economy whose development process is entirely driven by the spatio-temporal dynamics of capital accumulation. It allows to take into account spatial heterogeneities in technological level and population distribution. We solve analytically, via dynamic programming in infinite dimensions, the optimal control problem associated to the model, finding explicitly the optimal feedback and the value function. The expression of the optimal dynamics of the system in terms of eigenfunctions of an appropriate Sturm-Liouville problem allows to simulate the behavior of the variables and, in particular, their optimal discounted long-run spatial distribution.