Optimality Robustness in Koopman-Based Control

TL;DR

This paper investigates how uncertainties in Koopman-based control affect optimality, providing bounds, a robustness-aware control method, and a convergent policy iteration algorithm validated by numerical examples.

Contribution

It introduces a unified analysis framework for quantifying optimality deviations and proposes a robustness-enhanced control methodology with a convergent policy iteration algorithm.

Findings

Explicit bounds on deviations of value function and controller under uncertainties

A robustness-aware control method that reduces optimality deviations

A tractable policy iteration algorithm with proven convergence

Abstract

The Koopman operator enables simplified representations for nonlinear systems in data-driven optimal control, but the accompanying uncertainties inevitably induce deviations in the optimal controller and associated value function. This raises a distinct and fundamental question on optimality robustness, specifically, how uncertainties affect the optimal solution itself. To address this problem, we adopt a unified analysis-to-design perspective for systematically quantifying and improving optimality robustness. At the analysis level, we derive explicit upper bounds on the deviations of both the value function and the optimal controller, where uncertainties from multiple sources are systematically integrated into a unified norm-bounded representation. At the design level, we develop a robustness-aware optimal control methodology that provably reduces such optimality deviations, thereby enhancing robustness while explicitly revealing a quantitative trade-off between nominal optimality and robustness. As for practical implementation aspect, we further propose a tractable policy iteration algorithm, whose well-posedness and convergence are established via vanishing viscosity regularization and elliptic partial differential equation (PDE) techniques. Numerical examples validate the theoretical findings and demonstrate the effectiveness of proposed methodology.