A Unified Theory of Solvability for Stochastic Linear-Quadratic Control Problems I: The Homogeneous Case
Abstract
The existing theory of stochastic linear-quadratic (LQ) optimal control problems focuses on open-loop and closed-loop solvability. For finite horizon problems, it is widely accepted that those two types of solvability are not equivalent in general, as demonstrated by explicit counterexamples. In contrast, they coincide in standard infinite-horizon formulations with constant coefficients under suitable stabilizability and integrability assumptions. This paper aims at challenging this well-established paradigm by developing a novel framework of closed-loop solvability, namely generalized closed-loop solvability, which removes the L2-integrability constraint on the feedback strategies from the existing formulation. It will be proven that open-loop and generalized closed-loop solvability are equivalent and the classical closed-loop solvablity is a particular case only when the admissible feedback strategy is L2-integrable over the time horizon. In other words, the previously established non-equivalence between open-loop and closed-loop solvability stems solely from the limitations of the classical formulation which requires the L2-integrability of the admissible feedback strategies over the time-horizon. Therefore, this paper unifies the theory of the solvabilitiy for stochastic LQ control problems. Moreover, our results offer a novel insight to stochastic LQ control problems: open-loop/generalized closed-loop solvability is essentially equivalent to piecewisely local closed-loop solvability in the classical sense. The current paper is focused on homogeneous cases only and non-homogeneous cases will be studied in a subsequent paper.
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BibTeX
@misc{mei2026unified,
title = {{A Unified Theory of Solvability for Stochastic Linear-Quadratic Control Problems I: The Homogeneous Case}},
author = {Hongwei Mei},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27579},
eprint = {2610.27579},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27579},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - A Unified Theory of Solvability for Stochastic Linear-Quadratic Control Problems I: The Homogeneous Case
AU - Hongwei Mei
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27579
UR - https://arxiv.org/abs/2610.27579
AB - The existing theory of stochastic linear-quadratic (LQ) optimal control problems focuses on open-loop and closed-loop solvability. For finite horizon problems, it is widely accepted that those two types of solvability are not equivalent in general, as demonstrated by explicit counterexamples. In contrast, they coincide in standard infinite-horizon formulations with constant coefficients under suitable stabilizability and integrability assumptions. This paper aims at challenging this well-established paradigm by developing a novel framework of closed-loop solvability, namely generalized closed-loop solvability, which removes the L^2-integrability constraint on the feedback strategies from the existing formulation. It will be proven that open-loop and generalized closed-loop solvability are equivalent and the classical closed-loop solvablity is a particular case only when the admissible feedback strategy is L^2-integrable over the time horizon. In other words, the previously established non-equivalence between open-loop and closed-loop solvability stems solely from the limitations of the classical formulation which requires the L^2-integrability of the admissible feedback strategies over the time-horizon. Therefore, this paper unifies the theory of the solvabilitiy for stochastic LQ control problems. Moreover, our results offer a novel insight to stochastic LQ control problems: open-loop/generalized closed-loop solvability is essentially equivalent to piecewisely local closed-loop solvability in the classical sense. The current paper is focused on homogeneous cases only and non-homogeneous cases will be studied in a subsequent paper.
ER -