This talk consists of an overview of recent breakthroughs in solving nonlinear stochastic optimal control problems by leveraging on Koopman- and Fokker-Planck-Kolmogorov operators. The key idea of such operator-theoretic methods is to replace finite dimensional nonlinear closed-loop control systems by equivalent open-loop controlled linear systems in an infinite dimensional space. This enables the application of modern convex PDE-constrained optimization and control methods for solving nonlinear optimal control problems to global optimality. The talk discusses how to solve certain classes of nonlinear partial differential equations (PDEs), such as Hamilton-Jacobi-Bellman (HJB) equations, by using methods that have originally been designed for analyzing linear parabolic and elliptic PDEs. Moreover, a novel operator-theoretic dynamic programming recursion for bounded linear operators is introduced, which can be used to numerically solve HJB equations by a finite element method. The talks concludes with a summary of the far reaching consequences of these results for researchers working on nonlinear control theory as well as for developers of nonlinear optimal control algorithms.
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