Asymptotic problems in optimal control with a vanishing Lagrangian and unbounded data

TL;DR

This paper investigates the asymptotic behavior of finite horizon optimal control problems with unbounded data and nonnegative Lagrangians, providing a representation formula for the limit as the horizon extends to infinity.

Contribution

It introduces a new representation formula for the limit of finite horizon problems with unbounded data, linking it to the discounted problem and characterizing the limit via the HJB equation.

Findings

Derived a representation formula for the infinite horizon limit.

Established conditions for the limit to be the unique nonnegative solution of the HJB.

Discussed the connection to the ergodic control problem.

Abstract

In this paper we give a representation formula for the limit of the fnite horizon problem as the horizon becomes infinite, with a nonnegative Lagrangian and unbounded data. It is related to the limit of the discounted infinite horizon problem, as the discount factor goes to zero. We give sufficient conditions to characterize the limit function as unique nonnegative solution of the associated HJB equation. We also briefly discuss the ergodic problem.