arXiv:2610.41141v1 Announce Type: cross Abstract: While the Koopman operator-based lifted representations of nonlinear autonomous and controlled systems have been well covered in the literature, the direct controller synthesis problem largely remains open. As the controlled system is generally a bilinear (instead of fully linear) Koopman model, controller synthesis appears to lose the linear--quadratic structures of the classical Lyapunov and Riccati methods for finite-dimensional linear systems. However, in infinite dimensions, bilinearity in the feature map should be merely artifactual and hence reducible to linearity in principle. In this work, by considering the operator semigroups in the reproducing kernel Hilbert space with a linear--Sobolev kernel and utilizing the Yosida approximation of their generators, the synthesis problems of a stabilizing controller and an input-unconstrained optimal controller can be formulated as an operator Lyapunov inequality and an operator Riccati equation, respectively. These formulations recover the desirable linear--quadratic structures, and hence, with sample data, they are solvable efficiently in analogous ways to usual linear systems by semidefinite programming solvers and Kleinman iterations, respectively. In addition to proving control performance bounds dependent on the sampling denseness and Yosida regularization, through numerical comparison to linearized LQR, bilinear EDMD-based synthesis, nonlinear MPC, and Koopman MPC methods on numerical models and process control examples, this work demonstrates the proposed Koopman--Riccati synthesis as a computationally efficient approach to obtain a near-optimal state-feedback control law.