Linear error bounds for HJB equations in finite horizon control problems

TL;DR

This work improves error estimates for semi-Lagrangian approximations of finite-horizon Hamilton–Jacobi–Bellman equations by removing the pessimistic $1/\Delta t$-like term and proving a linear bound in $\Delta t$, $\Delta x$, and control oscillation $M_u$ under standard regularity and a positive discount $\lambda$. The authors extend infinite-horizon techniques to the finite-horizon setting, using a refined comparison of continuous and discrete costs and Grönwall-based stability to show $\|v-V\|_{\infty} \le C_1\Delta t + C_2\Delta x + C_3 M_u$. Numerical experiments with discounted and undiscounted problems confirm first-order convergence in space and time, consistent with the theoretical bound and suggesting robustness beyond the discounted case. The results enhance the reliability of SL schemes for practical finite-horizon control, and open avenues for extending the theory to $\lambda=0$ and other discretizations.

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 $\frac{1}{Δt}$ 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.

Figures (4)

• Figure 1: Test 1: Exact value function (left), optimal control (right).
• Figure 2: Test 1: Error convergence (left), approximated value function for $\Delta x = 0.00625$ (right).
• Figure 3: Test 2: Exact value function (left), approximate value function (right).
• Figure 4: Test 2: Experimental order of convergence for our example.

Theorems & Definitions (11)

• Theorem 2.1
• Remark
• Remark
• Theorem 3.1
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.2