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

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.

💡 Research Summary

The paper addresses optimal control problems for nonlinear ordinary differential equations (ODEs) whose right‑hand sides contain uncertain parameters. By applying the Sample Average Approximation (SAA) technique, the authors replace the expectation in the risk‑neutral objective with an empirical average over N independent realizations of the random parameter ξ. This transformation yields an “ensemble” optimal control problem consisting of N deterministic ODEs coupled only through the common control function u∈L²(0,1;ℝ^m).

Four main contributions are presented. (a) A non‑asymptotic mean convergence rate for the optimal values is proved: for every sample size N,
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