Convergence and Error Estimates of A Semi-Lagrangian scheme for the Minimum Time Problem

• Main Authors: Akian, Marianne, Liu, Shanqing
• Format: Journal Article
• Language: English
• Published: 09-07-2024
• Subjects: Mathematics - Optimization and Control
• DOI: 10.48550/arxiv.2407.06969

We consider a semi-Lagrangian scheme for solving the minimum time problem,
with a given target, and the associated eikonal type equation. We first use a
discrete time deterministic optimal control problem interpretation of the time
discretization scheme, and show that the discrete time value function is
semiconcave under regularity assumptions on the dynamics and the boundary of
target set. We establish a convergence rate of order $1$ in terms of time step
based on this semiconcavity property. Then, we use a discrete time stochastic
optimal control interpretation of the full discretization scheme, and we
establish a convergence rate of order $1$ in terms of both time and spatial
steps using certain interpolation operators, under further regularity
assumptions. We extend our convergence results to problems with particular
state constraints. We apply our results to analyze the convergence rate and
computational complexity of the fast-marching method. We also consider the
multi-level fast-marching method recently introduced by the authors.