Summary
Many important statements about dynamical systems can be proved by finding scalar-valued auxiliary functions whose time evolution along trajectories obeys certain pointwise inequalities that imply the desired result. The most familiar of these auxiliary functions is a Lyapunov function to prove steady-state stability, but such functions can also be used to bound averages of ergodic systems, define trapping boundaries, and so much more. In this talk I will highlight a method of identifying auxiliary functions from data using polynomial optimization. The method leverages recent advances in approximating the Koopman operator from data, so-called extended dynamic mode decomposition, to provide system-level information without system identification. The result is a flexible, data-driven, model-agnostic computational method that does not require explicit model discovery. Furthermore, it can be applied to data generated through a deterministic or stochastic process with no prior adjustments to the implementation. It can be used to bound quantities of interest, develop optimal state-dependent feedback controllers, and discover invariant measures.
