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For dynamical systems involving decision making, the success of the systemgreatly depends on its ability to make good decisions with incomplete anduncertain information. By leveraging the Koopman operator and its adjointproperty, we introduce the Koopman Expectation, an efficient method forcomputing expectations as propagated through a dynamical system. Unlike otherKoopman operator-based approaches in the literature, this is possible withoutan explicit representation of the Koopman operator. Furthermore, theefficiencies enabled by the Koopman Expectation are leveraged for optimizationunder uncertainty when expected losses and constraints are considered. We showhow the Koopman Expectation is applicable to discrete, continuous, and hybridnon-linear systems driven by process noise with non-Gaussian initial conditionand parametric uncertainties. We finish by demonstrating a 1700x accelerationfor calculating probabilistic quantities of a hybrid dynamical system over thenaive Monte Carlo approach with many orders of magnitudes improvement inaccuracy.

The Koopman Expectation: An Operator Theoretic Method for Efficient Analysis and Optimization of Uncertain Hybrid Dynamical Systems