A Tutorial on Pontryagin-Koopman Operators for Infinite Horizon Optimal Control

This paper provides a tutorial on how to use Koopman operators to lift Pontryagin's optimality condition for infinite-horizon optimal control problems into an infinite dimensional space. It is shown how to exploit the symplectic structure of the associated Pontryagin-Koopman operator in order to identify the stable manifold on which optimal trajectories evolve. Moreover, it is shown how to conduct a Koopman mode analysis in order to characterize optimal feedback control laws. Our focus is on providing a review of and gain further insight into the theory of Koopman-operator based optimal control methods. This is achieved by exploiting the structure of a particular optimal regulation problem, which is used as a tutorial example throughout the paper.