Remarks on the vanishing discount problem for infinite systems of Hamilton-Jacobi-Bellman equations

TL;DR

This paper investigates the behavior of solutions to infinite systems of Hamilton-Jacobi-Bellman equations as the discount factor approaches zero, establishing convergence and solvability of the related ergodic problem.

Contribution

It provides new results on the convergence and solvability of infinite coupled HJB systems in the vanishing discount limit.

Findings

Proves convergence of solutions as discount tends to zero

Establishes solvability of the ergodic problem for specific Hamiltonians

Analyzes asymptotic behavior of infinite HJB systems

Abstract

This paper is concerned with the asymptotic analysis of infinite systems of weakly coupled stationary Hamilton-Jacobi-Bellman equations as the discount factor tends to zero. With a specific Hamiltonian, we show the convergence of the solution and prove the solvability of the corresponding ergodic problem.