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We present a flexible data-driven method for dynamical system analysis thatdoes not require explicit model discovery. The method is rooted inwell-established techniques for approximating the Koopman operator from dataand is implemented as a semidefinite program that can be solved numerically.Furthermore, the method is agnostic of whether data is generated through adeterministic or stochastic process, so its implementation requires no prioradjustments by the user to accommodate these different scenarios. Rigorousconvergence results justify the applicability of the method, while alsoextending and uniting similar results from across the literature. Examples ondiscovering Lyapunov functions, performing ergodic optimization, and boundingextrema over attractors for both deterministic and stochastic dynamicsexemplify these convergence results and demonstrate the performance of themethod.

Auxiliary Functions as Koopman Observables: Data-Driven Analysis of Dynamical Systems via Polynomial Optimization