Time-invariant linear quadratic regulators: A new look
Abstract
This paper demonstrates that linear ordinary differential equations with constant coefficients serve as the main building block for time-invariant linear quadratic regulators (LQRs). By applying the Euler-Lagrange equations to a flat output—i.e., to a basis of the free finitely-generated module over the commutative principal ideal ring of time-invariant linear differential operators with respect to the time derivation—this result emerges directly. In module-theoretic terms, the freeness of this structure corresponds to the classical Kalman controllability condition (M. Fliess, Some basic structural properties of generalized linear systems, Systems and Control Letters, 15 (1990) 391-396). Within this framework, the two-point boundary value problem, including the control variables, becomes analytically tractable, enabling the derivation of optimal time horizons, optimal parameter design, and rest-to-rest trajectories. Furthermore, a novel turnpike-like phenomenon is established via nonstandard analysis. The feedback loop is closed thanks to the HEOL approach (C. Join, E. Delaleau, M. Fliess, Flatness-based control revisited: The HEOL setting, Comptes Rendus Math\'{e}matique, 362 (2024) 1693-1706), an extension of model-free control (MFC) that leverages intelligent controllers. Our viewpoint, which significantly simplifies the presentation by avoiding the conventional Kalman state-space representation, relies on foundational algebraic concepts that, while elementary, are rarely used in today's control theory. To aid intuition, some rudimentary examples are presented, most of which include computer simulations.
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BibTeX
@article{join2026timeinvariant,
title = {{Time-invariant linear quadratic regulators: A new look}},
author = {Cédric Join and Emmanuel Delaleau and Michel Fliess},
journal = {Linear Algebra and its Applications},
year = {2026},
month = oct,
doi = {10.1016/j.laa.2026.10.017},
eprint = {2512.10641},
archivePrefix = {arXiv},
url = {https://hal.sorbonne-universite.fr/hal-05828591v1},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Time-invariant linear quadratic regulators: A new look
AU - Cédric Join
AU - Emmanuel Delaleau
AU - Michel Fliess
PY - 2026
DA - 2026/10/13
JO - Linear Algebra and its Applications
DO - 10.1016/j.laa.2026.10.017
UR - https://hal.sorbonne-universite.fr/hal-05828591v1
AB - This paper demonstrates that linear ordinary differential equations with constant coefficients serve as the main building block for time-invariant linear quadratic regulators (LQRs). By applying the Euler-Lagrange equations to a flat output—i.e., to a basis of the free finitely-generated module over the commutative principal ideal ring of time-invariant linear differential operators with respect to the time derivation—this result emerges directly. In module-theoretic terms, the freeness of this structure corresponds to the classical Kalman controllability condition (M. Fliess, Some basic structural properties of generalized linear systems, Systems and Control Letters, 15 (1990) 391-396). Within this framework, the two-point boundary value problem, including the control variables, becomes analytically tractable, enabling the derivation of optimal time horizons, optimal parameter design, and rest-to-rest trajectories. Furthermore, a novel turnpike-like phenomenon is established via nonstandard analysis. The feedback loop is closed thanks to the HEOL approach (C. Join, E. Delaleau, M. Fliess, Flatness-based control revisited: The HEOL setting, Comptes Rendus Math\'{e}matique, 362 (2024) 1693-1706), an extension of model-free control (MFC) that leverages intelligent controllers. Our viewpoint, which significantly simplifies the presentation by avoiding the conventional Kalman state-space representation, relies on foundational algebraic concepts that, while elementary, are rarely used in today's control theory. To aid intuition, some rudimentary examples are presented, most of which include computer simulations.
SN - 0024-3795
SN - 1873-1856
ER -