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arxiv: 2604.05511 · v1 · submitted 2026-04-07 · 🧮 math.OC

A flatness proof of the exponential turnpike phenomenon for linear-quadratic optimal control problems

Michel Fliess (LJLL (UMR\_7598)) , Claude Lobry , Emmanuel Tr\'elat (LJLL (UMR\_7598) , CaGE) This is my paper

Pith reviewed 2026-05-10 19:29 UTC · model grok-4.3

classification 🧮 math.OC
keywords differential flatnessexponential turnpikelinear-quadratic optimal controlEuler-Lagrange equationstable-unstable splittingboundary layersSmith normal form
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The pith

For controllable linear-quadratic problems the exponential turnpike follows from stable-unstable splitting of the reduced flat Euler-Lagrange equation.

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

The paper shows that differential flatness supplies a direct proof of the exponential turnpike in finite-dimensional linear-quadratic optimal control. Controllability lets every trajectory be written in terms of a flat output and finitely many of its derivatives. Substituting this parametrization into the quadratic cost converts the Euler-Lagrange condition into a polynomial matrix differential equation. Reduction to Smith normal form turns the equation into scalar constant-coefficient problems whose solutions are exponential polynomials. When the reduced operator has no purely imaginary roots and the endpoint constraints act nondegenerately on the stable and unstable modes, the optimal trajectory therefore consists of left and right boundary layers separated by a long interior arc that stays exponentially close to the static optimum.

Core claim

If the pair (A, B) is controllable the linear control system is flat. Parametrizing trajectories by a flat output and its derivatives and inserting the parametrization into the quadratic functional produces a polynomial matrix differential equation. After reduction to Smith normal form this equation decouples into scalar constant-coefficient equations whose solutions are exponential polynomials. When the reduced Euler-Lagrange operator has no purely imaginary characteristic roots and when the endpoint constraints act nondegenerately on the stable and unstable modes, the optimal trajectory consists of a left-boundary layer, a right-boundary layer, and a long interior arc exponentially close<f

What carries the argument

The reduced flat equation obtained by substituting the flat-output parametrization into the quadratic cost and reducing the resulting Euler-Lagrange polynomial matrix differential equation to Smith normal form, allowing the stable-unstable splitting.

If this is right

  • For sufficiently long horizons the optimal trajectory exhibits distinct left and right boundary layers around an interior arc exponentially close to the static optimum.
  • When some weights are only semidefinite the order of the reduced equation drops and polynomial or oscillatory modes can appear, destroying the exponential turnpike.
  • Endpoint constraints involving the control and finitely many of its derivatives are well-defined on the smooth extremals selected by the reduced equation.
  • The same flatness reduction applies directly to the double-integrator example and produces the expected boundary-layer structure.

Where Pith is reading between the lines

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

  • The flat-output parametrization may supply an alternative route to constructing approximate turnpike solutions numerically by solving only the reduced scalar equations.
  • The stable-unstable splitting view could be tested on other flat linear systems to see whether the absence of imaginary roots is sufficient for the turnpike even without controllability.
  • When the reduced equation order drops, the method identifies which endpoint conditions become incompatible and therefore explains the precise manner in which the turnpike fails.

Load-bearing premise

The pair (A, B) is controllable and the endpoint constraints act nondegenerately on the stable and unstable modes of the reduced flat equation.

What would settle it

A concrete double-integrator calculation in which the reduced characteristic equation acquires a purely imaginary root or the endpoint constraints become degenerate on the stable-unstable modes, showing that the interior arc fails to stay exponentially close to the static optimum.

read the original abstract

We revisit finite-dimensional linear-quadratic optimal control from the viewpoint of differential flatness. If the pair (A, B) is controllable, then the linear control system is flat, and every trajectory can be parametrized by a flat output and finitely many of its derivatives. Once this parametrization is inserted into the quadratic functional, the Euler-Lagrange condition becomes a linear differential equation with constant coefficients, or more generally a polynomial matrix differential equation. After reduction to Smith normal form, this equation decouples into scalar constant-coefficient equations, and its solutions are exponentialpolynomials. This yields a viewpoint on the turnpike phenomenon that is quite different from the classical Hamiltonian-Riccati analysis: the turnpike mechanism appears directly from the stable-unstable splitting of the reduced flat equation. In particular, when the reduced Euler-Lagrange operator has no purely imaginary characteristic roots and when the endpoint constraints act nondegenerately on the stable and unstable modes, the optimal trajectory consists of a left-boundary layer, a right-boundary layer, and a long interior arc exponentially close to the static optimum. The same viewpoint also clarifies what changes when some weights are only semidefinite: the order of the reduced equation may drop, some endpoint conditions may become incompatible, and polynomial or oscillatory modes may destroy the exponential turnpike. It also gives a natural meaning to certain endpoint constraints on the control and on finitely many derivatives of the control: such traces are not defined on the ambient L 2 control space, but they are meaningful on the smooth extremals selected by the reduced Euler-Lagrange equation. We formulate this principle as a general theorem and illustrate it in detail on the double integrator.

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 proves the exponential turnpike phenomenon for finite-dimensional linear-quadratic optimal control problems via differential flatness. Assuming controllability of the pair (A, B), trajectories are parametrized by a flat output and its derivatives; substitution into the quadratic cost yields a constant-coefficient polynomial-matrix Euler-Lagrange equation. Smith normal form reduction decouples this into scalar equations whose solutions are exponential polynomials. Under the hypotheses that the reduced operator has no purely imaginary characteristic roots and that endpoint constraints act nondegenerately on the stable and unstable modes, the optimal trajectory decomposes into left and right boundary layers with a long interior arc exponentially close to the static optimum. The same framework clarifies the loss of exponential turnpike when weights are semidefinite (possible drop in equation order, incompatible endpoints, or polynomial/oscillatory modes) and gives meaning to endpoint constraints on the control and its derivatives. The general result is stated as a theorem and illustrated in detail on the double integrator.

Significance. If the central derivation holds, the manuscript supplies a genuinely new algebraic route to the turnpike property that bypasses the classical Hamiltonian-Riccati analysis and makes the stable-unstable splitting explicit after Smith reduction. The explicit handling of semidefinite weights, the interpretation of endpoint traces on the smooth extremals selected by the reduced equation, and the concrete double-integrator example are useful contributions. The approach is parameter-free once controllability and the stated nondegeneracy hypotheses are granted, and the caveats for the semidefinite case are clearly flagged.

minor comments (3)
  1. The precise statement of the nondegeneracy condition on the endpoint constraints (how they act on the stable and unstable modes after Smith reduction) should be isolated as a numbered hypothesis in the general theorem rather than appearing only in the surrounding text.
  2. In the double-integrator illustration, the explicit form of the reduced scalar Euler-Lagrange equation after Smith reduction and the resulting exponential-polynomial solutions should be written out with the characteristic roots labeled, to make the boundary-layer decay rates immediately visible.
  3. A short remark on how the flat-output parametrization extends (or fails to extend) when the control appears in the cost with a semidefinite weight would help readers see why the order drop occurs.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, accurate summary, and positive evaluation of the manuscript. The significance assessment and recommendation for minor revision are noted. We will incorporate any editorial improvements in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation is algebraic from controllability and flatness

full rationale

The paper starts from the standard controllability assumption implying differential flatness for linear systems, substitutes the flat output parametrization into the quadratic cost to obtain a constant-coefficient polynomial-matrix Euler-Lagrange equation, reduces it via Smith normal form to decoupled scalar equations, and extracts the turnpike from the stable-unstable splitting of the resulting exponential-polynomial solutions under the stated no-imaginary-roots and nondegeneracy hypotheses. This is a direct, self-contained mathematical reduction with explicit caveats for semidefinite weights; no parameters are fitted to data, no predictions are renamed fits, and no load-bearing step reduces to a self-citation or self-definition. The core claim follows from the algebraic structure and mode splitting without circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on controllability of (A,B) to guarantee flatness and on the absence of purely imaginary roots plus nondegeneracy of endpoint constraints; these are standard domain assumptions in linear systems theory rather than ad-hoc inventions.

axioms (3)
  • domain assumption The pair (A, B) is controllable
    Invoked in the abstract to ensure the system is flat and every trajectory can be parametrized by a flat output.
  • domain assumption The reduced Euler-Lagrange operator has no purely imaginary characteristic roots
    Required for the stable-unstable splitting to produce exponential decay to the static optimum.
  • domain assumption Endpoint constraints act nondegenerately on stable and unstable modes
    Needed so that the boundary layers can be chosen to satisfy the constraints without destroying the interior turnpike arc.

pith-pipeline@v0.9.0 · 5636 in / 1569 out tokens · 54166 ms · 2026-05-10T19:29:49.092988+00:00 · methodology

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