Optimality Robustness in Koopman-Based Control

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 naturally raises the question of how such uncertainty-induced optimality deviation can be quantified and mitigated. To address this problem, we adopt a unified analysis-to-design perspective that connects the characterization of optimality robustness with its improvement through controller design. At the analysis level, we establish a unified treatment of multiple uncertainty sources in Koopman-based control, where approximation error and noisy data are incorporated into a common robustness analysis through a 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.