Asymptotic analysis for Hamilton-Jacobi-Bellman equations on Euclidean space

TL;DR

This paper investigates the long-term behavior of solutions to Hamilton-Jacobi-Bellman equations on Euclidean space, especially when the Hamiltonian is non-Tonelli, establishing convergence, existence of solutions, and their relation to control systems.

Contribution

It extends asymptotic analysis of HJB equations to non-Tonelli Hamiltonians on Euclidean space, proving convergence and existence of solutions under Lie algebra conditions.

Findings

Proves convergence of the time-averaged value function as horizon approaches infinity.

Establishes existence of solutions to the ergodic equation under Lie algebra rank condition.

Constructs a critical solution coinciding with Lax-Oleinik evolution.

Abstract

The long-time average behavior of the value function in the calculus of variations is known to be connected to the existence of the limit of the corresponding Abel means. Still in the Tonelli case, such a limit is in turn related to the existence of solutions of the critical Hamilton-Jacobi equation. The goal of this paper is to address similar issues when set on the whole Euclidean space and the Hamiltonian fails to be Tonelli. We first study the convergence of the time-averaged value function as the time horizon goes to infinity, proving the existence of the critical constant for a general control system. Then, we show that the ergodic equation admits solutions for systems associated with a family of vector fields which satisfies the Lie Algebra rank condition. Finally, we construct a critical solution of the HJB equation on the whole space which coincides with its Lax-Oleinik evolution.