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

TL;DR

This work targets risk-neutral optimal control problems with uncertain inputs in affine-control ODEs and solves them via a sample-average approximation that yields an ensemble of deterministic dynamics. By leveraging metric-entropy bounds and gradient regularity in Hilbert spaces, it proves nonasymptotic, Monte Carlo-type convergence rates for both the SAA optimal values and the SAA-criticality measures, and establishes a uniform bound for Hilbert-space-valued sub-Gaussian Carathéodory mappings. The authors provide explicit rate formulas, quantify the需 sample complexity through covering-number bounds, and validate the theory on two numerical examples—the harmonic oscillator and an epidemic vaccination-scheduling problem under parameter uncertainty. The results offer finite-sample guarantees for SAA in infinite-dimensional control with uncertainty, guiding sample-size choices and informing robust control design in applications.

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.

Figures (4)

• Figure 1: For the harmonic oscillator problem formulated in \ref{['sec:harmonicoscillator']}, nominal solution (left) and reference solution (right).
• Figure 2: For the harmonic oscillator problem formulated in \ref{['sec:harmonicoscillator']}, convergence rate of the SAA optimal values (left) and of the reference criticality measure evaluated at SAA critical points (right). The empirical means $\widehat{\mathbb{E}}$ were computed using $50$ replications. The convergence rates were computed using least squares.
• Figure 3: For the vaccination scheduling problem formulated in \ref{['subsec:vaccination']}, nominal solution (left) and reference solution (right).
• Figure 4: For the vaccination scheduling problem formulated in \ref{['subsec:vaccination']}, convergence rate of the SAA optimal values as computed by the optimization solver (left) and of the reference criticality measure evaluated at SAA critical points (right). The empirical means $\widehat{\mathbb{E}}$ were computed using $50$ replications. The convergence rates were computed using least squares.

Theorems & Definitions (25)

• Remark 3.1
• Theorem 4.1
• Lemma 4.2
• proof
• Lemma 4.3
• proof
• proof : Proof of \ref{['thm:gradientregularity']}
• Theorem 5.1
• proof
• Theorem 6.1