On the accessibility and controllability of statistical linearization for stochastic control: algebraic rank conditions and their genericity

Statistical linearization has recently seen a particular surge of in-terest as a numerically cheap method for robust control of stochastic differential equations. Although it has already been successfully applied to control com-plex stochastic systems, accessibility and controllability properties of statisti-cal linearization, which are key to make the robust control problem well-posed, have not been investigated yet. In this paper, we bridge this gap by providing sufficient conditions for the accessibility and controllability of statistical lin-earization. Specifically, we establish simple sufficient algebraic conditions for the accessibility and controllability of statistical linearization, which involve the rank of the Lie algebra generated by the drift only. In addition, we show these latter algebraic conditions are essentially sharp, by means of a coun-terexample, and that they are generic with respect to the drift and the initial condition.