On the Accessibility and Controllability of Statistical Linearization for Stochastic Control: Algebraic Rank Conditions and their Genericity

TL;DR

This paper investigates the accessibility and controllability of statistical linearization in stochastic control, providing algebraic rank conditions and demonstrating their genericity, which are crucial for ensuring well-posedness of control problems.

Contribution

It introduces simple algebraic rank conditions for controllability and accessibility of statistical linearization, linking them to the Lie algebra rank of the drift, and proves their genericity.

Findings

Provided algebraic conditions for controllability and accessibility.

Showed these conditions are sharp via a counterexample.

Proved the conditions are generic with respect to drift and initial state.

Abstract

Statistical linearization has recently seen a particular surge of interest as a numerically cheap method for robust control of stochastic differential equations. Although it has already been successfully applied to control complex stochastic systems, accessibility and controllability properties of statistical 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 linearization. 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 counterexample, and that they are generic with respect to the drift and the initial condition.