Optimal bounds for numerical approximations of finite horizon problems based on dynamic programming approach

TL;DR

This paper establishes optimal error bounds for fully discrete dynamic programming approximations of finite horizon control problems, achieving first-order convergence in both time and space under standard regularity assumptions.

Contribution

It extends previous infinite horizon error analysis to finite horizon problems, providing optimal bounds and avoiding restrictive control regularity assumptions.

Findings

Proves an $O(h+k)$ error bound for the method

Achieves first-order convergence in time and space

Avoids restrictive regularity assumptions on controls

Abstract

In this paper we provide optimal bounds for fully discrete approximations to finite horizon problems via dynamic programming. We adapt the error analysis in \cite{nos} for the infinite horizon case to the finite horizon case. We prove an a priori bound of size $O(h+k)$ for the method, $h$ being the time discretization step and $k$ the spatial mesh size. Arguing with piecewise constants controls we are able to obtain first order of convergence in time and space under standard regularity assumptions, avoiding the more restrictive regularity assumptions on the controls required in \cite{nos}. We show that the loss in the rate of convergence in time of the infinite case (obtained arguing with piece-wise controls) can be avoided in the finite horizon case