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arxiv: 2604.22487 · v1 · submitted 2026-04-24 · 🧮 math.OC

Non static exponential turnpike property for optimal control problems with symmetries and boundary conditions

Sofya Maslovskaya , Sina Ober-Bl\"obaum , Boris Wembe This is my paper

Pith reviewed 2026-05-08 11:10 UTC · model grok-4.3

classification 🧮 math.OC
keywords optimal controlturnpike propertyLie group symmetriesHamiltonian reductionboundary value problemstrim primitivesexponential convergenceAbelian symmetries
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The pith

Optimal trajectories with Abelian symmetries stay exponentially close to a trim primitive except near the boundaries under hyperbolicity of the reduced equilibrium.

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

The paper proves an exponential trim turnpike property for optimal control problems whose dynamics and costs admit structural properties coming from Abelian Lie group symmetries. By constructing a reduced optimal control problem, the authors show that the original extremals are exactly the solutions of a reduced Hamiltonian boundary value problem whose static version defines a trim primitive. When the equilibrium of that reduced Hamiltonian system is hyperbolic, optimal trajectories of the original problem remain exponentially close to this trim primitive for long time horizons, with all deviations confined to thin boundary layers at the initial and final times. A reader would care because the result replaces the need to solve a full dynamic boundary value problem by the far simpler task of solving the static reduced problem and adding local corrections only near the endpoints.

Core claim

For optimal control problems possessing structural properties related to Abelian Lie group symmetries, the authors introduce a reduced optimal control problem whose optimality system coincides with the reduced Hamiltonian boundary value problem characterizing the original extremals. Under a hyperbolicity assumption on the equilibrium of the corresponding reduced Hamiltonian system, they prove that optimal trajectories remain exponentially close, up to boundary layers near the endpoints, to a trim primitive defined by the static reduced problem.

What carries the argument

The reduced optimal control problem obtained by quotienting out Abelian Lie group symmetries, whose static version supplies the trim primitive and whose Hamiltonian boundary value problem recovers all extremals of the original problem.

If this is right

  • For sufficiently long horizons the bulk of any optimal trajectory can be replaced by the trim primitive coming from the static reduced problem.
  • Initial and terminal boundary layers of exponentially decaying thickness absorb the mismatch between the trim and the prescribed boundary conditions.
  • The same reduction and hyperbolicity argument applies uniformly to linear quadratic problems, nonlinear quadratic problems, and the Kepler orbital transfer problem.
  • The resulting turnpike is intrinsically non-stationary because it follows the symmetry-induced trim trajectory rather than a fixed point.

Where Pith is reading between the lines

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

  • The same reduction could be applied numerically by first solving the static reduced problem and then using its trim as a warm start for the full dynamic solver with boundary corrections.
  • If hyperbolicity holds only approximately, one would still expect practical exponential closeness on finite but long intervals, which could be verified by direct simulation on the given examples.
  • The technique suggests that turnpike-type results may hold on other symmetry classes once an analogous static reduction is available.

Load-bearing premise

The equilibrium of the reduced Hamiltonian system must be hyperbolic, so that its stable and unstable manifolds produce exponential attraction to the trim primitive away from the boundaries.

What would settle it

A concrete optimal control problem satisfying the Abelian symmetry and structural assumptions, with a hyperbolic reduced equilibrium, for which numerical solution of a long-horizon instance shows that the distance from the optimal trajectory to the trim primitive does not decay exponentially in the interior of the time interval.

read the original abstract

Optimal control problems with symmetries often admit a non stationary turnpike property called trim turnpike, which characterizes the convergence of optimal solutions to certain symmetry induced trajectories called trim primitives. In this paper we establish an exponential trim turnpike property for a class of optimal control problems with structural properties related to Abelian Lie group symmetries. The key ingredient of our approach is the introduction of an appropriate reduced optimal control problem. We show that extremals of the original problem can be characterized through a reduced Hamiltonian boundary value problem that coincides with the optimality system of the reduced problem. Under a hyperbolicity assumption on the equilibrium of the corresponding reduced Hamiltonian system we prove that optimal trajectories remain exponentially close, up to boundary layers near the endpoints, to a trim primitive defined by the static reduced problem. The theoretical results are illustrated on three representative examples: linear and nonlinear problems with quadratic cost and the Kepler orbital transfer problem.

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

2 major / 2 minor

Summary. The paper establishes an exponential trim turnpike property for optimal control problems with Abelian Lie group symmetries. It introduces a reduced optimal control problem such that the extremals of the original problem are characterized by a reduced Hamiltonian boundary value problem coinciding with the optimality system of the reduced problem. Under a hyperbolicity assumption on the equilibrium of the reduced Hamiltonian system, optimal trajectories remain exponentially close to a trim primitive (defined by the static reduced problem), up to boundary layers near the endpoints. The results are illustrated on three examples: linear and nonlinear problems with quadratic cost, and the Kepler orbital transfer problem.

Significance. If the derivations hold, the work extends turnpike theory to non-stationary symmetric settings by leveraging Abelian symmetry reduction to obtain a static reduced problem whose optimality system matches the original extremals. This provides a conditional but general framework for problems with symmetries, such as orbital mechanics. The explicit hyperbolicity assumption and the three illustrative examples (including a nonlinear application) are strengths that make the contribution concrete and applicable.

major comments (2)
  1. [Reduction procedure] The reduction step: the manuscript asserts that the reduced Hamiltonian BVP coincides exactly with the optimality system of the reduced problem (including boundary conditions). The derivation of this coincidence from the Abelian Lie group symmetries should be expanded with explicit steps showing how the original extremals map to the reduced ones, as this is load-bearing for the entire trim turnpike claim.
  2. [Examples] Hyperbolicity assumption: while the exponential convergence is proved under this assumption on the reduced Hamiltonian equilibrium, the paper should verify or provide conditions ensuring the assumption holds for the Kepler example (and the nonlinear quadratic case), since the trim primitive and boundary-layer behavior depend directly on it.
minor comments (2)
  1. Clarify the precise definition of 'trim primitive' early in the introduction, as it is central to the non-static turnpike statement.
  2. In the abstract and title, ensure consistent use of 'non-static' versus 'non static' for readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and recommendation of minor revision. We address the two major comments point by point below and will revise the manuscript to incorporate the requested clarifications and verifications.

read point-by-point responses

  1. Referee: [Reduction procedure] The reduction step: the manuscript asserts that the reduced Hamiltonian BVP coincides exactly with the optimality system of the reduced problem (including boundary conditions). The derivation of this coincidence from the Abelian Lie group symmetries should be expanded with explicit steps showing how the original extremals map to the reduced ones, as this is load-bearing for the entire trim turnpike claim.

    Authors: We agree that the reduction procedure is central and that a more explicit derivation will strengthen the exposition. In the revised manuscript we will expand the relevant section with a detailed, step-by-step account: we will first recall the momentum map associated with the Abelian action, then show how the original Pontryagin extremals (state, costate, and control) are mapped to the reduced variables, and finally verify that the reduced Hamiltonian boundary-value problem, including the boundary conditions, coincides exactly with the first-order optimality conditions of the reduced optimal control problem. This will make the load-bearing step fully transparent. revision: yes

  2. Referee: [Examples] Hyperbolicity assumption: while the exponential convergence is proved under this assumption on the reduced Hamiltonian equilibrium, the paper should verify or provide conditions ensuring the assumption holds for the Kepler example (and the nonlinear quadratic case), since the trim primitive and boundary-layer behavior depend directly on it.

    Authors: We appreciate the suggestion. In the revised examples section we will add explicit verification or sufficient conditions. For the nonlinear quadratic-cost problem we will state a simple algebraic condition on the cost matrices that guarantees hyperbolicity of the reduced equilibrium. For the Kepler orbital-transfer example we will compute (or bound) the eigenvalues of the linearized reduced Hamiltonian vector field at the trim primitive and confirm that none lie on the imaginary axis for the physical parameters used in the numerical illustration. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper defines a reduced optimal control problem via Abelian Lie group symmetries, establishes that the original extremals coincide exactly with the reduced optimality system via the reduction, and then invokes an external hyperbolicity assumption on the reduced Hamiltonian equilibrium to prove exponential closeness to the trim primitive (up to boundary layers). This structure is a standard symmetry-reduction technique in optimal control; the central claim is conditional on explicitly stated assumptions that are independent of the target result, with no reduction of any prediction to fitted inputs, self-definitional loops, or load-bearing self-citations. The three examples serve as illustrations rather than verification that would create circularity. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests primarily on the hyperbolicity assumption for the reduced system and the existence of a reduction enabled by Abelian symmetries; no free parameters or invented entities are mentioned.

axioms (2)
  • domain assumption Hyperbolicity of the equilibrium of the reduced Hamiltonian system
    Required to guarantee exponential closeness to the trim primitive away from boundaries.
  • domain assumption Structural properties related to Abelian Lie group symmetries allow reduction to a static problem whose optimality system matches the original
    This enables the characterization of extremals via the reduced Hamiltonian boundary value problem.

pith-pipeline@v0.9.0 · 5457 in / 1413 out tokens · 67130 ms · 2026-05-08T11:10:32.951460+00:00 · methodology

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