A semi-Lagrangian scheme for Hamilton-Jacobi-Bellman equations with oblique boundary conditions

TL;DR

This paper develops a fully-discrete semi-Lagrangian scheme for second order Hamilton-Jacobi-Bellman equations with oblique boundary conditions, proving convergence to the unique viscosity solution and confirming results with numerical tests.

Contribution

It introduces a novel semi-Lagrangian scheme for HJB equations with oblique boundary conditions, ensuring consistency, monotonicity, stability, and convergence.

Findings

Scheme is consistent, monotone, and stable.

Proven convergence to the viscosity solution.

Numerical results confirm theoretical convergence.

Abstract

We investigate in this work a fully-discrete semi-Lagrangian approximation of second order possibly degenerate Hamilton-Jacobi-Bellman (HJB) equations on a bounded domain with oblique boundary conditions. These equations appear naturally in the study of optimal control of diffusion processes with oblique reflection at the boundary of the domain. The proposed scheme is shown to satisfy a consistency type property, it is monotone and stable. Our main result is the convergence of the numerical solution towards the unique viscosity solution of the HJB equation. The convergence result holds under the same asymptotic relation between the time and space discretization steps as in the classical setting for semi-Lagrangian schemes. We present some numerical results that confirm the numerical convergence of the scheme.