In this paper, we study how the Koopman operator framework can be combined with kernel methods to effectively control nonlinear dynamical systems. While kernel methods have typically large computational requirements, we show how random subspaces (Nystr\"om approximation) can be used to achieve huge computational savings while preserving accuracy. Our main technical contribution is deriving theoretical guarantees on the effect of the Nystr\"om approximation. More precisely, we study the linear quadratic regulator problem, showing that the approximated Riccati operator converges at the rate $m^{-1/2}$, and the regulator objective, for the associated solution of the optimal control problem, converges at the rate $m^{-1}$, where $m$ is the random subspace size. Theoretical findings are complemented by numerical experiments corroborating our results.

翻译：本文研究了如何将Koopman算子框架与核方法相结合，以实现对非线性动力系统的有效控制。尽管核方法通常具有较高的计算需求，但我们证明了如何利用随机子空间（Nyström近似）在保持精度的同时实现巨大的计算节省。我们的主要技术贡献在于推导了Nyström近似效果的理论保证。具体而言，我们研究了线性二次调节器问题，证明了近似Riccati算子以$m^{-1/2}$的速率收敛，且最优控制问题对应解的调节目标函数以$m^{-1}$的速率收敛，其中$m$为随机子空间维度。理论结果辅以数值实验，进一步验证了我们的结论。