Dynamics and control of an integro-differential system of geographical economics

In this paper we consider the impact of induced environmental pollution on the qualitative behavior and control of a system of geographical economics. Our underlying mathematical model extends other results in the literature along different directions. A general class of production functions is considered, including, in addition to the classical Cobb-Douglas production function, convex-concave production functions. The dynamics of the pollution is modelled via a diffusion equation coupled, via an integral source, with the geographically distributed production. Reciprocally, we suppose that the (negative) influence of pollution may be modeled as a negative feedback acting on the production function, and therefore on capital accumulation. We analyze the qualitative behavior of the coupled system, andthen propose an optimal control problem for the above model. In order to solve the system of partial differential equations which describes the optimality conditions, we implement a Forward-Backward Sweep algorithm. Numerical simulations are reported which illustrate the behavior of the system and its optimal control.