Koopman--Nemytskii Operator of Nonlinear Controlled Systems and Its Learning for Controller Synthesis

While the Koopman operator represents a nonlinear system as a linear operator in a function space, its definition does not involve inputs. For controller synthesis, an operator model is needed to describe the effect of feedback laws on closed-loop systems, so that the desired state-feedback law can be computationally searched based on such a predictive model. To this end, this paper proposes a Koopman--Nemytskii operator, defined as a linear operator that maps canonical features of state--policy pairs in a reproducing kernel Hilbert space (RKHS) to that of succeeding states. Under regularity conditions on the dynamics and kernel selection, this operator is definable on suitable Sobolev-type RKHSs, and its data-based estimation guarantees bounded errors in single-step prediction, multi-step prediction, and accumulated cost under control. The controller synthesis problem is thus formulated as a convex kernel-based optimization one and efficiently solved in a sample-based manner.