Minimal norm control problems for linear mean-field control systems
Abstract
This paper studies controllability and minimal norm control problems (NP problems) for mean-field stochastic differential equation (MF-SDE) systems. We first establish an equivalent rank criterion for terminal controllability, namely, that the diffusion matrices D and D+D̅ are both of full row rank. We then derive an explicit Riccati-based representation of the minimal norm control by applying a parameterization technique. A key step is to identify the value P(0) of an associated Riccati solution with the controllability Gramian of a controllable auxiliary system, which yields the parameter required in the construction. The resulting representation provides a basis for the future development and analysis of numerical methods for NP problems.
Related
- Minimal control time of some 1D first-order linear hyperbolic systems with row rank deficient boundary coupling matrix
- The Well-Posedness and Attainability of an LQ Optimal Control Problem for Mean-Field Linear Stochastic Systems Controlled by Impulses
- Minimal null control time of some 1D linear hyperbolic balance laws with constant coefficients and properties of related kernel equations
- Numerical approximation of distributed optimal control problems governed by linear thermo-poroelasticity
- Connection between MP and DPP for Mean-Field Stochastic Optimal Control Problems
BibTeX
@misc{an2026minimal,
title = {{Minimal norm control problems for linear mean-field control systems}},
author = {Manfen An and Rui Gao and Yanqing Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17825},
eprint = {2610.17825},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17825},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Minimal norm control problems for linear mean-field control systems
AU - Manfen An
AU - Rui Gao
AU - Yanqing Wang
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17825
UR - https://arxiv.org/abs/2610.17825
AB - This paper studies controllability and minimal norm control problems (NP problems) for mean-field stochastic differential equation (MF-SDE) systems. We first establish an equivalent rank criterion for terminal controllability, namely, that the diffusion matrices D and D+D̅ are both of full row rank. We then derive an explicit Riccati-based representation of the minimal norm control by applying a parameterization technique. A key step is to identify the value P(0) of an associated Riccati solution with the controllability Gramian of a controllable auxiliary system, which yields the parameter required in the construction. The resulting representation provides a basis for the future development and analysis of numerical methods for NP problems.
ER -