Extending Koopman operator theory to control systems requires accounting for the fundamentally different roles of states and inputs. We introduce the Koopman Control Family (KCF), a rigorous operator-theoretic framework for general discrete-time nonlinear control systems, without assuming a control-affine structure. The KCF consists of the Koopman operators associated with all input-free systems induced by setting the input constant. We show that this family fully characterizes the control system. Although the KCF may contain uncountably many operators, the entire family can be parametrized through a single Koopman operator on an augmented function space.
For finite-dimensional computation, we derive a universal finite-dimensional form, termed the “input-state separable” form, which includes commonly used linear, bilinear, and switched-linear models as special cases. We construct optimal finite-dimensional approximations on inner-product function spaces and quantify their worst-case prediction error. These results provide accuracy guarantees and a principled foundation for robust data-driven learning and control.
Masih Haseli is a Postdoctoral Scholar Research Associate in Computing and Mathematical Sciences at the California Institute of Technology. He received the Ph.D. degree in Engineering Sciences (Mechanical Engineering) from the University of California, San Diego, in 2022. From 2022 to 2025, he was a postdoctoral scholar in the Department of Mechanical and Aerospace Engineering at UC San Diego. His research develops theoretically principled, data-driven methods for modeling, analysis, and control of nonlinear dynamical systems, with applications in learning and robotics. He is the recipient of the 2025 IEEE Control Systems Letters Outstanding Paper Award and the Best Student Paper Award at the 2021 American Control Conference.
