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Understanding the stability and long-term behavior of dynamical systems is vital in numerous applications. These properties can be studied through Lyapunov frameworks that, when the dynamics are governed by known polynomial models, can be implemented computationally using polynomial optimization. But what if the model is not polynomial or, worse, not even known?
In this talk, I will show that polynomial optimization can be combined with extended dynamic mode decomposition to perform system analysis directly from measured data. This is possible thanks to a connection between Lyapunov frameworks and the Koopman operator. After introducing the basic theory behind this connection, I will show how data-driven system analysis can guarantee stability and unravel chaotic dynamics on a range of examples.