The Koopman-Hill projection method offers an efficient approach for stability analysis of linear time-periodic systems, and thereby also for the Floquet stability analysis of periodic solutions of nonlinear systems. However, its accuracy has previously been supported only by numerical evidence, lacking rigorous theoretical guarantees. This paper presents the first explicit error bound for the truncation error of the Koopman-Hill projection method, establishing a solid theoretical foundation for its application. The bound applies to linear time-periodic systems whose Fourier coefficients decay exponentially with a sufficient rate, and is derived using constructive series expansions. The bound quantifies the difference between the true and approximated fundamental solution matrices, clarifies conditions for guaranteed convergence, and enables conservative but reliable inference of Floquet multipliers and stability properties. Additionally, the same methodology applied to a subharmonic formulation demonstrates improved convergence rates of the latter. Numerical examples, including the Mathieu equation and the Duffing oscillator, illustrate the practical relevance of the bound and underscore its importance as the first rigorous theoretical justification for the Koopman-Hill projection method.

翻译：Koopman-Hill投影方法为线性时周期系统的稳定性分析提供了一种高效途径，从而也可用于非线性系统周期解的Floquet稳定性分析。然而，其准确性此前仅得到数值证据的支持，缺乏严格的理论保证。本文首次给出了Koopman-Hill投影方法截断误差的显式误差界，为其应用奠定了坚实的理论基础。该误差界适用于傅里叶系数以足够速率指数衰减的线性时周期系统，并通过构造性级数展开推导得出。该界量化了真实基本解矩阵与近似基本解矩阵之间的差异，阐明了保证收敛的条件，并能够对Floquet乘子及稳定性性质进行保守但可靠的推断。此外，将相同方法应用于次谐波形式，证明了后者具有更快的收敛速率。数值算例（包括Mathieu方程和Duffing振子）验证了该误差界的实际相关性，并强调了其作为Koopman-Hill投影方法首个严格理论证明的重要性。