Dynamic programming for infinite horizon boundary control problems of PDE's with age structure

TL;DR

This paper develops a dynamic programming framework for infinite horizon boundary control problems involving PDEs with age structure, proving the uniqueness of the value function and feedback controls, with applications to economic models of vintage capital.

Contribution

It extends finite horizon results to the infinite horizon case for PDE boundary control problems, providing existence and uniqueness of solutions and controls.

Findings

Value function is the unique regular solution of the stationary HJB equation.

Existence and uniqueness of feedback controls are established.

The approach is motivated by and applicable to economic models of optimal investment with vintage capital.

Abstract

We develop the dynamic programming approach for a family of infinite horizon boundary control problems with linear state equation and convex cost. We prove that the value function of the problem is the unique regular solution of the associated stationary Hamilton--Jacobi--Bellman equation and use this to prove existence and uniqueness of feedback controls. The idea of studying this kind of problem comes from economic applications, in particular from models of optimal investment with vintage capital. Such family of problems has already been studied in the finite horizon case by Faggian. The infinite horizon case is more difficult to treat and it is more interesting from the point of view of economic applications, where what mainly matters is the behavior of optimal trajectories and controls in the long run. The study of infinite horizon is here performed through a nontrivial limiting procedure from the corresponding finite horizon problem.