  We introduce a spatial economic growth model where space is described as a
network of interconnected geographic locations and we study a corresponding
finite-dimensional optimal control problem on a graph with state constraints.
Economic growth models on networks are motivated by the nature of spatial
economic data, which naturally possess a graph-like structure: this fact makes
these models well-suited for numerical implementation and calibration. The
network setting is different from the one adopted in the related literature,
where space is modeled as a subset of a Euclidean space, which gives rise to
infinite dimensional optimal control problems. After introducing the model and
the related control problem, we prove existence and uniqueness of an optimal
control and a regularity result for the value function, which sets up the basis
for a deeper study of the optimal strategies. Then, we focus on specific cases
where it is possible to find, under suitable assumptions, an explicit solution
of the control problem. Finally, we discuss the cases of networks of two and
three geographic locations.
