On a parabolic Hamilton-Jacobi-Bellman equation degenerating at the boundary

TL;DR

This paper studies the long-term behavior of solutions to a boundary-degenerate Hamilton-Jacobi-Bellman equation, showing convergence to a steady state without boundary conditions under certain degeneracy assumptions.

Contribution

It establishes the asymptotic convergence of solutions to a steady state for a boundary-degenerate HJB equation, extending ergodic problem analysis to new degeneracy conditions.

Findings

Solutions converge uniformly to a steady state

No boundary data needed due to degeneracy condition

Existence of a constant velocity c for convergence

Abstract

We derive the long time asymptotic of solutions to an evolutive Hamilton-Jacobi-Bellman equation in a bounded smooth domain, in connection with ergodic problems recently studied in \cite{bcr}. Our main assumption is an appropriate degeneracy condition on the operator at the boundary. This condition is related to the characteristic boundary points for linear operators as well as to the irrelevant points for the generalized Dirichlet problem, and implies in particular that no boundary datum has to be imposed. We prove that there exists a constant $c$ such that the solutions of the evolutive problem converge uniformly, in the reference frame moving with constant velocity $c$, to a unique steady state solving a suitable ergodic problem.