Exact Controllability of Backward-Structured Mean-Field SDEs: Hautus Criteria and Initial-Time Dichotomy
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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BibTeX
@misc{sun2026exact,
title = {{Exact Controllability of Backward-Structured Mean-Field SDEs: Hautus Criteria and Initial-Time Dichotomy}},
author = {Jingrui Sun and Lvning Yuan and Xurun Zuo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23072},
eprint = {2609.23072},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23072},
}
RIS
TY - UNPB
TI - Exact Controllability of Backward-Structured Mean-Field SDEs: Hautus Criteria and Initial-Time Dichotomy
AU - Jingrui Sun
AU - Lvning Yuan
AU - Xurun Zuo
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.23072
UR - https://arxiv.org/abs/2609.23072
AB - 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.
ER -