Data-driven system analysis: Polynomial optimization meets Koopman

Questions about stability, long-time behaviour, and the effect of uncertainty are at the heart of nonlinear system analysis. Many of these questions can be answered using "auxiliary functions", which generalize the Lyapunov functions encountered in stability analysis. Moreover, auxiliary functions can be constructed using polynomial optimization if a system is governed by known polynomial equations. But what if the governing equations are not polynomial or, worse, are not known?

In this talk, I will show that auxiliary functions can be "discovered" directly from data if one combines polynomial optimization with (extended) dynamic mode decomposition. This enables one to perform data-driven system analysis without having to first identify a model for the system dynamics. The key to this is a previously unrecognized connection between auxiliary functions and the Koopman operator, which can be approximated from data with rigorous convergence guarantees. After explaining this connection, I will present examples illustrating how the method allows one to perform stability analysis, bound long-time average behaviour, and extract unstable periodic orbits for low-dimensional chaotic attractors.

All results in this talk were obtained in collaboration with Jason Bramburger (Concordia University).