Event description
In this talk, I will introduce a family of optimal control problems with state constraints, characterized by linear state dynamics and a concave gain functional.
This is motivated by economic growth problems with a spatial dimension, whose aim is to understand the relationship between capital accumulation and spatial agglomeration in a given region, taking into account the geographic distribution of the relevant economic variables.
These models consider the point of view of a centralized social planner, who aims at maximizing an aggregate intertemporal utility of consumption, to determine the optimal capital and consumption paths at each time and each geographic location. I will discuss both the cases where geography is described as a continuum of locations (e.g., the unit circle) or as a network of distinct locations (e.g. a finite, simple graph).
In this framework, state dynamics are expressed either by a Partial Differential Equation, or by a system of Ordinary Differental Equations.
In both cases, standard results cannot be applied, due to the peculiar structure of the problem under study.
I will show some results on the Hamilton-Jacobi-Bellman equation associated to the optimal control problem, and on the regularity of the value function. Then, I will provide suitable conditions under which it is possible to explicitly find the optimal control. Finally, I will analyze the long-run behaviour of the so-called detrended optimal capital paths and the stability of the corresponding steady states.
This is joint work with S. Federico, F. Gozzi, M. Leocata, G. I. Papagiannis, A. Xepapadeas, A. N. Yannacopoulos.