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

Javier de Frutos, Julia Novo

TL;DR

Addresses sharp a priori error bounds for fully discrete semi-Lagrangian schemes solving finite-horizon Hamilton-Jacobi-Bellman problems. The authors discretize time with Euler steps and space with piecewise-linear interpolation, achieving an overall convergence rate of $O(h+k)$ under standard assumptions and a convexity condition. They prove a bound of the form $|v_{h,k}^n(x)-v(x,t_n)| \le C_1 h + C_2 k$, with $C_2$ reflecting a discrete Lipschitz constant of nodal controls, and show how horizon finiteness avoids a time-convergence degradation seen in the infinite-horizon case. The work provides a rigorous framework for error control in finite-horizon DP-based feedback synthesis and lays groundwork for reduced-order extensions such as POD.

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

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

TL;DR

Addresses sharp a priori error bounds for fully discrete semi-Lagrangian schemes solving finite-horizon Hamilton-Jacobi-Bellman problems. The authors discretize time with Euler steps and space with piecewise-linear interpolation, achieving an overall convergence rate of under standard assumptions and a convexity condition. They prove a bound of the form , with reflecting a discrete Lipschitz constant of nodal controls, and show how horizon finiteness avoids a time-convergence degradation seen in the infinite-horizon case. The work provides a rigorous framework for error control in finite-horizon DP-based feedback synthesis and lays groundwork for reduced-order extensions such as POD.

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 for the method, being the time discretization step and 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
Paper Structure (5 sections, 5 theorems, 100 equations)

This paper contains 5 sections, 5 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Model problem and preliminary results
  3. Fully discrete approximation
  4. Error analysis of the method
  5. Interpolation bounds

Key Result

Theorem 1

For any $x\in \overline \Omega$, $n=0,\ldots,N$, the function $w_{h,k}^n(x)$ satifies the equation and, as a consequence, for each node $x_i$, $i=1,\dots,n_s$

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Proof 1
  • Theorem 3
  • Proof 2
  • Lemma 2
  • Proof 3