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This paper is about operator-theoretic methods for solving nonlinearstochastic optimal control problems to global optimality. These methodsleverage on the convex duality between optimally controlled diffusion processesand Hamilton-Jacobi-Bellman (HJB) equations for nonlinear systems in an ergodicHilbert-Sobolev space. In detail, a generalized Bakry-Emery condition isintroduced under which one can establish the global exponential stabilizabilityof a large class of nonlinear systems. It is shown that this condition issufficient to ensure the existence of solutions of the ergodic HJB forstochastic optimal control problems on infinite time horizons. Moreover, anovel dynamic programming recursion for bounded linear operators is introduced,which can be used to numerically solve HJB equations by a Galerkin projection.

Convex operator-theoretic methods in stochastic control