Convergence rates for ensemble-based solutions to optimal control of uncertain dynamical systems

TL;DR

This paper develops convergence rate analysis for ensemble-based optimal control solutions of uncertain nonlinear dynamical systems, validated through numerical simulations on physical and epidemiological models.

Contribution

It introduces a theoretical framework providing non-asymptotic convergence rates for ensemble-based control methods using metric entropy bounds.

Findings

Convergence rates are established for ensemble solutions.

Numerical validation on harmonic oscillator and epidemic models.

Framework applicable to uncertain nonlinear control problems.

Abstract

We consider optimal control problems involving nonlinear ordinary differential equations with uncertain inputs. Using the sample average approximation, we obtain optimal control problems with ensembles of deterministic dynamical systems. Leveraging techniques for metric entropy bounds, we derive non-asymptotic Monte Carlo-type convergence rates for the ensemble-based solutions. Our theoretical framework is validated through numerical simulations on a harmonic oscillator problem and a vaccination scheduling problem for epidemic control under model parameter uncertainty.