On a mean-field Pontryagin minimum principle for stochastic optimal control

Abstract: This papers outlines a novel extension of the classical Pontryagin minimum (maximum) principle to stochastic optimal control problems. Contrary to the well-known stochastic Pontryagin minimum principle involving forward-backward stochastic differential equations, the proposed formulation is deterministic and of mean-field type. The Hamiltonian structure of the proposed Pontryagin minimum principle is achieved via the introduction of an appropriate gauge variable. The gauge freedom can be used to decouple the forward and reverse time equations; hence simplifying the solution of the underlying boundary value problem. We also consider infinite horizon discounted cost optimal control problems. In this case, the mean-field formulation allows converting the computation of the desired optimal control law into solving a pair of forward mean-field ordinary differential equations. The proposed mean-field formulation of the Pontryagin minimum principle is tested numerically for a controlled inverted pendulum and a controlled Lorenz-63 system.