State constrained control problems in Banach lattices and applications

TL;DR

This paper develops a novel framework for solving deterministic optimal control problems with positivity constraints in infinite dimensional Banach lattices, providing explicit solutions and feedback controls, especially in economic applications.

Contribution

It introduces a new approach to control problems in Banach lattices with positivity constraints, including explicit solutions to HJB equations and optimal feedback controls, extending previous work beyond L^2 spaces.

Findings

Explicit solutions to HJB equations in Banach lattices

Derivation of optimal feedback controls in infinite dimensions

Application of Perron-Frobenius theorem to control paths

Abstract

This paper aims to study a family of deterministic optimal control problems in infinite dimensional spaces. The peculiar feature of such problems is the presence of a positivity state constraint, which often arises in economic applications. To deal with such constraints, we set up the problem in a Banach space with a Riesz space structure (i.e., a Banach lattice) not necessarily reflexive: a typical example is the space of continuous functions on a compact set. In this setting, which seems to be new in this context, we are able to find explicit solutions to the Hamilton-Jacobi-Bellman (HJB) equation associated to a suitable auxiliary problem and to write the corresponding optimal feedback control. Thanks to a type of infinite dimensional Perron-Frobenius Theorem, we use these results to get information about the optimal paths of the original problem. This was not possible in the infinite dimensional setting used in earlier works on this subject, where the state space was an $\mathrm L^2$ space.