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arXiv · 2609.23072

Exact Controllability of Backward-Structured Mean-Field SDEs: Hautus Criteria and Initial-Time Dichotomy

Jingrui Sun·Lvning Yuan·Xurun Zuo
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Abstract

This paper studies exact controllability of linear mean-field stochastic differential equations with backward-structure. We establish Hautus criteria and uncover a sharp dichotomy between the zero initial time and positive initial times. At time zero, exact controllability is characterized by a Hautus condition for the mean dynamics, with additional control directions generated by the centered stochastic dynamics. At any positive initial time, exact controllability further requires a stochastic Hautus condition for the centered system. Unlike the classical eigenvector-based condition, this stochastic criterion is formulated in terms of positive-semidefinite eigenmatrices of an associated Lyapunov-type operator. The dichotomy arises from the triviality of the initial sigma-field. We also clarify the controllability relations among the corresponding ODE, SDE, and mean-field SDE systems.

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Jingrui Sun, Lvning Yuan, Xurun Zuo. 2026-09-19. Exact Controllability of Backward-Structured Mean-Field SDEs: Hautus Criteria and Initial-Time Dichotomy. https://arxiv.org/abs/2609.23072

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