Spatially Aware Dictionary-Free Koopman Eigenfunction Identification for Modeling and Control

A spatially aware dictionary-free eigenfunction discovery (SADFED) framework is proposed for identification of low-rank Koopman models from data without prescribing a lifting dictionary, kernel, or neural-network eigenfunction architecture. A reference trajectory is selected and used to determine the Koopman modes by regularized least squares (LS). Then, a transformed temporal basis allows the eigenfunction values at all sampled initial conditions to be obtained by a second regularized LS projection. Consequently, only the real and imaginary parts of the eigenvalues remain as the optimization variables. Interpolation of the identified eigenfunction samples reveals their spatial structure, enabling numerical estimation of their gradients. A joint objective combines trajectory reconstruction error with a normalized Koopman partial differential equation (KPDE) residual, promoting spatial consistency with the KPDE over the sampled region and serving as a physics-informed regularizer. The method is evaluated on a system with analytical Koopman eigenfunctions, the FitzHugh-Nagumo system, the van der Pol oscillator, the Duffing system, and a two-spool turbojet engine. The examples demonstrate recovery of known eigenfunctions, sensitivity to reference trajectory and hyperparameters, limit-cycle harmonics and isochrons, discontinuous indicator eigenfunctions and isostables, symmetry exploitation, and construction of state-dependent lifted input dynamics. For the turbojet example, the identified model is further used for state estimation and design of a gain-scheduled tracking linear quadratic Gaussian controller. The results indicate the applicability of SADFED to Koopman spectral identification and control-oriented modeling of nonlinear dynamical systems.