Hongwei Mei · 2026 · arXiv (Cornell University)
DOI: 10.48550/arxiv.2610.27579 · cited by 0 works (OpenAlex) · open-access PDF
A teoria existente é a de que os problemas de controle óptimo de circuito fechado linear-quadrado (LQ) estão focados em circuito aberto e circuito fechado. Para problemas de horizonte finito, é amplamente aceito que esses dois tipos de solvabilidade não são equivalentes em geral, como demonstrado por contra-exemplos explícitos. Em contraste, eles coincidem em formulações de horizonte infinito com coeficientes constantes sob pressupostos de estabilização e integração adequados. Este artigo visa desafiar este paradigma de feedback bem estabelecido, desenvolvendo um novo quadro de solvabilidade de circuito fechado, ou seja, solvabilidade de circuito fechado generalizada, que remove as restrições de gravidade infinita padrão de $L2$ sobre as estratégias de feedback contínuo. Será provado que o controle de
Machine translation (Meta M2M100); the English original follows.
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.
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Topics: Mathematics, Constraint (computer-aided design), Constant (computer programming), Homogeneous, Control (management), Applied mathematics
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