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arxiv: 2604.04003 · v1 · submitted 2026-04-05 · 🧮 math.OC

A Periodic Dichotomy in Linear Control Theory

Shichao Ye , Xingwu Zeng , Can Zhang This is my paper

Pith reviewed 2026-05-13 17:19 UTC · model grok-4.3

classification 🧮 math.OC
keywords periodic dichotomyRiccati equationLyapunov equationlinear quadratic controloptimal extremalstabilizabilitydetectabilityperiodic systems
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The pith

A periodic dichotomy built from Riccati and Lyapunov solutions gives an explicit form for the optimal extremal in periodic linear-quadratic control.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors construct a periodic dichotomy transformation by combining solutions of periodic Riccati and Lyapunov equations. This transformation is then used to derive an explicit representation of the optimal extremal trajectory for periodic linear quadratic problems. Under the assumptions of exponential stabilizability and detectability, the construction yields a complete characterization of that extremal. A reader would care because periodic systems appear in applications with repeating cycles, and an explicit form removes the need to solve the two-point boundary-value problem numerically at each step.

Core claim

We construct a periodic dichotomy transformation using solutions of periodic Riccati and Lyapunov equations. As an application of this transformation, we provide an explicit representation of the optimal extremal for periodic linear quadratic optimal control problems. Specifically, we establish a complete characterization of the optimal extremal under suitable exponential stabilizability and detectability assumptions.

What carries the argument

The periodic dichotomy transformation, which decomposes the state space into stable and unstable invariant subspaces via the periodic Riccati and Lyapunov solutions.

If this is right

  • The optimal trajectory is fully determined once the stable and unstable periodic subspaces are obtained from the Riccati and Lyapunov solutions.
  • Periodic linear-quadratic problems admit closed-form extremals rather than requiring iterative numerical solution of the two-point boundary-value problem.
  • The dichotomy supplies the precise boundary conditions that separate the finite-cost trajectories from the infinite-cost ones.
  • Any periodic feedback that achieves the optimal cost must coincide with the one induced by the stable subspace of the dichotomy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Numerical schemes for periodic Riccati equations could be used as a practical route to compute the optimal extremal without discretizing the entire horizon.
  • The same splitting may extend to finding explicit solutions for periodic Kalman filtering or periodic H-infinity problems.
  • If the period tends to zero the construction should recover the classical constant-coefficient dichotomy used in infinite-horizon LQ theory.

Load-bearing premise

The linear system must be exponentially stabilizable and detectable so that the required periodic Riccati and Lyapunov solutions exist and define the dichotomy.

What would settle it

A concrete periodic linear system that is exponentially stabilizable and detectable yet whose optimal extremal trajectory fails to satisfy the explicit representation given by the constructed dichotomy.

Figures

Figures reproduced from arXiv: 2604.04003 by Can Zhang, Shichao Ye, Xingwu Zeng.

Figure 3
Figure 3. Figure 3: Matrix P(·) 0 2 4 6 t -0.6 -0.5 -0.4 -0.3 -0.2 E11 0 2 4 6 t -0.2 -0.15 -0.1 -0.05 0 E12 0 2 4 6 t -0.2 -0.15 -0.1 -0.05 0 E21 0 2 4 6 t -0.6 -0.5 -0.4 -0.3 -0.2 E22 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 1
Figure 1. Figure 1: Two distinct periodic orbits of P11 Let P11(·) denote the (1, 1)-entry of the solution to (5). We observe empirically that when (5) is integrated backward from the final time with an arbitrary initial terminal state, P11(·) converges asymptotically to a stable periodic orbit. This is illustrated in Figure 1a, obtained using the MATLAB solver ode45. Conversely, when integrated forward in time, the solution … view at source ↗
Figure 2
Figure 2. Figure 2: Characteristic multipliers (eigenvalues) of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 5
Figure 5. Figure 5: Periodic optimal extremal (yθ, λθ, uθ) exponentially close to a periodic orbit associated with a periodic LQ optimal control problem. This property holds identically for the optimal control and the adjoint state. We consider the following optimal control problem, denoted by (LQ) T . For any T > 0, consider the linear control system ( y˙(t) = A(t)y(t) + B(t)u(t), t ∈ (0, T), y(0) = y0, and the optimal contr… view at source ↗
Figure 6
Figure 6. Figure 6: Turnpike phenomenon As depicted in [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
read the original abstract

In this paper, we construct a periodic dichotomy transformation using solutions of periodic Riccati and Lyapunov equations. As an application of this transformation, we provide an explicit representation of the optimal extremal for periodic linear quadratic optimal control problems. Specifically, we establish a complete characterization of the optimal extremal under suitable exponential stabilizability and detectability assumptions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper constructs a periodic dichotomy transformation from solutions of the periodic Riccati and Lyapunov equations and applies it to obtain an explicit representation of the optimal extremal trajectory for periodic linear-quadratic optimal control problems. The central result is a complete characterization of this extremal under the standard assumptions of exponential stabilizability and detectability.

Significance. If the construction is correct, the result supplies a clean, explicit description of the optimal trajectory that directly leverages the unique positive-semidefinite periodic Riccati solution guaranteed by the stabilizability-detectability hypotheses. This extends the classical dichotomy transformation technique to the periodic setting without introducing new fitting parameters or hidden regularity conditions on the period, and it yields a falsifiable representation that can be checked against the monodromy matrix of the closed-loop system.

minor comments (3)
  1. [Abstract] The abstract states the construction but does not indicate the precise section where the transformation matrix is first defined; adding a forward reference (e.g., “see §3.2”) would improve readability.
  2. [§2] Notation for the periodic Riccati solution P(t) and the associated dichotomy matrix should be introduced once in the preliminaries and used consistently thereafter; occasional reuse of the symbol X for both the state and the transformation risks confusion.
  3. [Theorem 4.1] The statement of Theorem 4.1 claims the representation is “parameter-free”; a brief remark confirming that the only data entering the transformation are the given periodic coefficients A(t), B(t), Q(t), R(t) would strengthen this claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the positive assessment. We are grateful for the recommendation to accept the paper.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The central construction defines the periodic dichotomy transformation explicitly from solutions of the standard periodic Riccati and Lyapunov differential equations. These equations are independently posed in the periodic linear systems literature and exist under the paper's stated exponential stabilizability and detectability assumptions, which are classical sufficient conditions and do not presuppose the target optimal-extremal characterization. No step reduces a claimed prediction or uniqueness result to a fitted quantity inside the paper, nor does any load-bearing premise rest on a self-citation whose content is itself unverified. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claim rests on the standard domain assumption that exponential stabilizability and detectability imply existence of periodic Riccati and Lyapunov solutions; the dichotomy itself is the main new object introduced.

axioms (1)
  • domain assumption Exponential stabilizability and detectability imply existence of stabilizing periodic Riccati and Lyapunov solutions
    Invoked to guarantee the dichotomy transformation exists and is well-defined.
invented entities (1)
  • Periodic dichotomy transformation no independent evidence
    purpose: To decompose the state space into forward- and backward-stable periodic subspaces for explicit optimal control representation
    New object constructed in the paper from the equation solutions; no independent falsifiable prediction supplied in the abstract.

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Works this paper leans on

15 extracted references · 15 canonical work pages

  1. [1]

    Tracking periodic signals with the overtaking criterion,

    Z. Artstein and A. Leizarowitz, “Tracking periodic signals with the overtaking criterion,” IEEE Trans. Autom. Control , vol. 30, no. 11, pp. 1123–1126, Nov. 1985

    work page 1985

  2. [2]

    Analysis of the periodic Lyapunov and Riccati equations via canonical decomposition,

    S. Bittanti, P. Colaneri, and G. Guardabassi, “Analysis of the periodic Lyapunov and Riccati equations via canonical decomposition,” SIAM J. Control Optim., vol. 24, pp. 1138-1149, 1986

    work page 1986

  3. [3]

    The recursive reduced-order numerical solution of the singularly perturbed matrix differential Riccati equation,

    T. Grodt and Z. Gajic, “The recursive reduced-order numerical solution of the singularly perturbed matrix differential Riccati equation,” IEEE Trans. Autom. Control, vol. 33, pp. 751-754, 1988

    work page 1988

  4. [4]

    A numerical evaluation of solvers for the periodic Riccati differential equation,

    S. Gusev, S. Johansson, B. K ˚agstr¨om, A. Shiriaev, and A. Varga, “A numerical evaluation of solvers for the periodic Riccati differential equation,” BIT Numer. Math., vol. 50, no. 2, pp. 301–329, Jun. 2010. 11

    work page 2010

  5. [5]

    Liberzon, Calculus of Variations and Optimal Control Theory: A Concise Introduction

    D. Liberzon, Calculus of Variations and Optimal Control Theory: A Concise Introduction. Princeton, NJ, USA: Princeton University Press, 2012

    work page 2012

  6. [6]

    Long time versus steady state optimal control,

    A. Porretta and E. Zuazua, “Long time versus steady state optimal control,” SIAM J. Control Optim. , vol. 51, no. 6, pp. 4242–4273, 2013

    work page 2013

  7. [7]

    A. S. Poznyak, Advanced Mathematical Tools for Automatic Control En- gineers: Deterministic Techniques, vol. 1. Amsterdam, The Netherlands: Elsevier, 2008

    work page 2008

  8. [8]

    Dichotomic basis approach to solving hyper-sensitive optimal control problems,

    A. V . Rao and K. D. Mease, “Dichotomic basis approach to solving hyper-sensitive optimal control problems,” Automatica, vol. 35, pp. 633- 642, 1999

    work page 1999

  9. [9]

    The exponential turnpike property for periodic linear quadratic optimal control problems in infinite dimension,

    E. Tr ´elat, X. Zeng, and C. Zhang, “The exponential turnpike property for periodic linear quadratic optimal control problems in infinite dimension,” SIAM J. Control Optim. , vol. 63, no. 4, pp. 2524–2546, 2025

    work page 2025

  10. [10]

    Steady-state and periodic exponen- tial turnpike property for optimal control problems in Hilbert spaces,

    E. Tr ´elat, C. Zhang, and E. Zuazua, “Steady-state and periodic exponen- tial turnpike property for optimal control problems in Hilbert spaces,” SIAM J. Control Optim. , vol. 56, pp. 1222-1252, 2018

    work page 2018

  11. [11]

    The turnpike property in finite-dimensional nonlinear optimal control,

    E. Tr ´elat and E. Zuazua, “The turnpike property in finite-dimensional nonlinear optimal control,” J. Differential Equations , vol. 258, pp. 81- 114, 2015

    work page 2015

  12. [12]

    On solving periodic Riccati equations,

    A. Varga, “On solving periodic Riccati equations,” Numer. Linear Algebra Appl., vol. 15, no. 9, pp. 809–835, 2008

    work page 2008

  13. [13]

    A dichotomy in linear control theory,

    R. Wilde and P. Kokotovic, “A dichotomy in linear control theory,” IEEE Trans. Autom. Control, vol. 17, pp. 382-383, 1972

    work page 1972

  14. [14]

    Wang and Y

    G. Wang and Y . Xu, Periodic Feedback Stabilization for Linear Periodic Evolution Equations , SpringerBriefs in Mathematics. Cham, Switzer- land: Springer, 2017

    work page 2017

  15. [15]

    Characterization by detectability inequality for periodic stabi- lization of linear time-periodic evolution systems,

    Y . Xu, “Characterization by detectability inequality for periodic stabi- lization of linear time-periodic evolution systems,”Systems Control Lett., vol. 149, pp. 104871, Jan. 2021

    work page 2021