Linear error bounds for HJB equations in finite horizon control problems

TL;DR

This paper improves error bounds for numerical schemes solving Hamilton-Jacobi-Bellman equations in finite horizon control, showing linear dependence on discretization parameters and confirming first-order convergence through experiments.

Contribution

It provides sharper, linear error estimates for semi-Lagrangian schemes under standard assumptions, removing previous pessimistic bounds involving inverse time steps.

Findings

Error bounds depend linearly on time step and mesh size

Numerical experiments confirm first-order convergence

Improved bounds hold even without discounting

Abstract

We study semi Lagrangian approximation schemes for Hamilton Jacobi Bellman equations arising from finite horizon optimal control problems. Classical error estimates for these schemes include the term $\frac{1}{\Delta t}$ which leads to pessimistic convergence bounds and is not observed in numerical experiments. In this work, we provide improved error estimates under standard regularity assumptions on the dynamics, the running cost, and the final cost, assuming the presence of a positive discount factor. The new bound depends linearly on the time step, the spatial mesh size, and a measure of the temporal oscillation of the control, thus removing the mixed term appearing in previous analyses. The proof relies on a refined comparison between continuous and discrete cost functionals and on stability estimates for the controlled dynamics. Numerical experiments confirm first-order convergence in both space and time and suggest that the improved behavior persists even in the undiscounted case.