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 both the approximated Riccati operator and the regulator objective, for the associated solution of the optimal control problem, converge at the rate $m^{-1/2}$, where $m$ is the random subspace size. Theoretical findings are complemented by numerical experiments corroborating our results.

翻译：本文研究如何将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大的计算开销，但我们展示了如何利用随机子空间（Nyström近似）在保持精度的同时大幅降低计算成本。本文的主要技术贡献在于推导了Nyström近似影响的理论保证。具体而言，我们针对线性二次型调节器问题，证明了近似Riccati算子以及最优控制问题相关解的调节器目标均以速率$m^{-1/2}$收敛，其中$m$为随机子空间大小。数值实验进一步佐证了理论结果的正确性。