On the Existence of Quadratic Control Lyapunov Functions for Koopman-Operator based Bilinear Systems

TL;DR

This paper investigates the limitations of quadratic control Lyapunov functions for Koopman-based bilinear systems, showing their existence implies constant input stabilizability, and proposes a convex relaxation for validation.

Contribution

It establishes a link between quadratic CLFs and stabilizability in Koopman bilinear systems and introduces a semidefinite relaxation for practical validation.

Findings

Quadratic CLFs imply the system can be stabilized by a constant input.

The proposed SDP relaxation provides a sufficient condition for CLF validity.

Empirical results suggest the approach extends to high-dimensional multi-input systems.

Abstract

Koopman operator-based methods enable data-driven bilinear representations of unknown nonlinear control systems. Accurate representations often demand significantly higher dimensions than the original system, making control design challenging. Control Lyapunov Functions (CLFs) are widely used for controller synthesis, with quadratic CLF candidates being the most common due to their simplicity. Yet, we show that this class is highly restrictive, especially when the state dimension is large: under mild conditions, their existence implies stabilizability of the bilinear system by a constant input -- that is, the control remains fixed over time. We establish this result by formulating a quadratically constrained quadratic program (QCQP) that exactly characterizes valid CLFs. Since QCQPs are NP-hard, we propose a convex semidefinite relaxation that offers a sufficient validity condition. For single-input systems, we prove that a quadratic CLF requires constant control stabilizability, and empirically demonstrate that this extends to high-dimensional multi-input systems in many cases.