Optimality Deviation using the Koopman Operator

This paper investigates the impact of approximation error in data-driven optimal control problem of nonlinear systems while using the Koopman operator. While the Koopman operator enables a simplified representation of nonlinear dynamics through a lifted state space, the presence of approximation error inevitably leads to deviations in the computed optimal controller and the resulting value function. We derive explicit upper bounds for these optimality deviations, which characterize the worst-case effect of approximation error. Supported by numerical examples, these theoretical findings provide a quantitative foundation for improving the robustness of data-driven optimal controller design.

💡 Research Summary

This paper addresses a critical gap in data‑driven optimal control of nonlinear systems using the Koopman operator: the quantitative effect of approximation errors on the resulting optimal controller and value function. The authors consider a control‑affine nonlinear system (\dot x = f(x) + \sum_{i=1}^m g_i(x)u_i) and lift it to a higher‑dimensional space via a dictionary of observables (\Psi(x)). In the lifted coordinates (z = \Psi(x)) the dynamics become a bilinear system (\dot z = Az + B_0 u + \sum_{i=1}^m B_i z u_i) plus an error term (r(z,u)) that captures both projection error (finite‑dimensional lifting) and estimation error (finite data).

A key modeling assumption is that the error is bounded linearly in the state and input norms: (|r(z,u)| \le c_1|z| + c_2|u|) for known constants (c_1, c_2 > 0). This assumption is justified by recent probabilistic and deterministic error‑bound results for EDMD/extended DMD and related kernel methods.

The infinite‑horizon quadratic cost \