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arxiv: 2605.12467 · v1 · submitted 2026-05-12 · 📡 eess.SY · cs.SY

Recognition: 2 theorem links

· Lean Theorem

Towards Closed-loop Stability of Nonlinear Receding Horizon Games

Florian D\"orfler, Sophie Hall, Timm Faulwasser

Pith reviewed 2026-05-13 03:26 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords receding horizon gamesturnpike phenomenonrecursive feasibilityclosed-loop stabilitygeneralized Nash equilibriumnonlinear systemsmodel predictive controlasymptotic convergence
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The pith

Receding horizon games for nonlinear systems achieve recursive feasibility and practical stability from the turnpike property without needing terminal ingredients.

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

The paper establishes that in receding horizon games—multi-agent optimization over a finite horizon without terminal costs or constraints—recursive feasibility can be inferred from the turnpike phenomenon under mild assumptions on the system dynamics and cost functions. This leads to sufficient conditions for practical asymptotic convergence of closed-loop trajectories to the steady-state generalized Nash equilibrium. The authors demonstrate through numerical examples that the region of attraction around this equilibrium shrinks exponentially with increasing horizon length, a known trait in model predictive control, and show that a simple linear end penalty can suppress undesired leaving arcs to achieve asymptotic convergence.

Core claim

We show that recursive feasibility can be inferred from the turnpike phenomenon under mild assumptions. Moreover, we prove sufficient conditions for practical asymptotic convergence of the closed-loop trajectories, and we discuss how the gap towards practical asymptotic stability may be closed. Numerical examples illustrate that the closed-loop region of attraction around the steady-state GNE shrinks exponentially with the horizon length.

What carries the argument

The turnpike phenomenon, where optimal trajectories remain close to the steady-state equilibrium for most of the horizon, which is used to infer recursive feasibility and stability without terminal ingredients.

If this is right

  • Recursive feasibility holds without any terminal sets or costs.
  • Closed-loop trajectories exhibit practical asymptotic convergence to the GNE.
  • The basin of attraction decreases exponentially as the prediction horizon lengthens.
  • A linear terminal penalty can ensure asymptotic rather than practical stability.

Where Pith is reading between the lines

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

  • This approach could simplify the design of stable multi-agent control systems by avoiding the need for complex terminal constraints.
  • It suggests that turnpike-based arguments might apply to other horizon-based game-theoretic control problems.
  • Extending the linear penalty idea might provide a general way to bridge practical to asymptotic stability in similar settings.

Load-bearing premise

The turnpike phenomenon must hold for the nonlinear dynamics and cost functions under the given mild assumptions.

What would settle it

A numerical simulation or analytical example where the turnpike property is satisfied but the closed-loop system becomes infeasible or diverges from the GNE for some initial conditions.

Figures

Figures reproduced from arXiv: 2605.12467 by Florian D\"orfler, Sophie Hall, Timm Faulwasser.

Figure 1
Figure 1. Figure 1: A schematic showing the turnpike property of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A schematic of the balls B around xs used in Lemma 2, Lemma 3, and Theorem 2. Hence whenever ∥x ∗ 0|t − xs∥ > α−1 ℓ (P¯ + nLℓ(ρ(ε) + ∆(ε))) we have that ∆W < 0 and thus also W(xt+1) − W(xt) < 0, ∀xt ̸∈ Bρ˜(xs) with ρ˜ := α −1 ℓ [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 5
Figure 5. Figure 5: Closed-loop trajectories (black-solid) and open-loop [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Convergence of closed-loop trajectories of ( [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 4
Figure 4. Figure 4: Convergence to xs of closed-loop GNE trajectories of (18) after T = 20 control steps. with v ∈ {1, 2}. The parameter values are A = 1.5, B1 = 1, B2 = 2, R1,1 = R1,2 = 4, R2,2 = R2,1 = 5 and state weights Q1 = 1, Q2 = 2. The reference state is x ref = 0.3. We solve for GNEs of (18) using a regularized Fischer– Burmeister method [40]. We show the resulting trajectories in [PITH_FULL_IMAGE:figures/full_fig_p… view at source ↗
read the original abstract

We analyze Receding Horizon Games without any MPC-like terminal ingredients. We show that recursive feasibility can be inferred from the turnpike phenomenon under mild assumptions. Moreover, we prove sufficient conditions for practical asymptotic convergence of the closed-loop trajectories, and we discuss how the gap towards practical asymptotic stability may be closed. We use numerical examples to show that the closed-loop region of attraction around the steady-state GNE shrinks exponentially with the horizon length, a behavior previously known only for model predictive control. Further, we apply a linear end penalty and demonstrate in numerical simulations that it suppresses the leaving arc and ensures asymptotic convergence to the steady-state GNE.

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 analyzes receding horizon games for nonlinear dynamical systems without MPC-style terminal ingredients. It claims that recursive feasibility of the finite-horizon game can be inferred directly from the turnpike property under mild assumptions on the dynamics and stage costs. Sufficient conditions are derived for practical asymptotic convergence of the closed-loop trajectories to the steady-state generalized Nash equilibrium (GNE), with discussion of how the remaining gap to practical asymptotic stability can be closed. Numerical examples are used to demonstrate that the region of attraction around the steady-state GNE shrinks exponentially with the prediction horizon and that a linear end penalty suppresses the leaving arc and restores asymptotic convergence.

Significance. If the central claims hold, the work provides a terminal-ingredient-free stability framework for receding-horizon games that parallels existing MPC turnpike results, with concrete numerical evidence of exponential region-of-attraction shrinkage and the practical benefit of a linear end penalty. The explicit linkage of recursive feasibility to the turnpike phenomenon and the derivation of practical convergence conditions represent a useful extension of single-agent MPC theory to multi-agent settings.

major comments (2)
  1. [Abstract and main theoretical section] The central feasibility argument rests on the external turnpike property holding under the stated mild assumptions; however, the manuscript does not appear to contain an explicit verification or counter-example check that the turnpike property is indeed satisfied for the class of nonlinear dynamics and costs considered (see the abstract claim and the discussion following the main theorem).
  2. [Theoretical results section] The sufficient conditions for practical asymptotic convergence are stated to be proved in the manuscript, yet the provided abstract and structure do not include the explicit assumption list, error-bound derivations, or the precise statement of the convergence theorem that would allow verification of whether the practical-stability gap is rigorously characterized.
minor comments (2)
  1. Notation for the steady-state GNE and the leaving arc should be introduced earlier and used consistently throughout the numerical examples.
  2. The numerical examples would benefit from an explicit statement of the system dimensions, cost parameters, and horizon lengths used to generate the exponential-shrinkage plots.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed review of our manuscript. We address each major comment point by point below, providing clarifications from the paper and indicating the revisions we will make to improve clarity and verifiability.

read point-by-point responses

  1. Referee: [Abstract and main theoretical section] The central feasibility argument rests on the external turnpike property holding under the stated mild assumptions; however, the manuscript does not appear to contain an explicit verification or counter-example check that the turnpike property is indeed satisfied for the class of nonlinear dynamics and costs considered (see the abstract claim and the discussion following the main theorem).

    Authors: We thank the referee for highlighting this point. The manuscript invokes the external turnpike property under the mild assumptions on the nonlinear dynamics and stage costs (explicitly listed as Assumption 1 in Section II), from which recursive feasibility of the finite-horizon game is inferred directly, as shown in the proof of the main feasibility result (Theorem 1). The turnpike property itself is not re-proved from first principles but is taken as holding for the considered class, consistent with standard practice in turnpike-based MPC literature. To address the concern and make the argument more self-contained, we will add a short verification remark in the revised Section III, including a brief check that the assumptions are satisfied for the dynamics and costs in our numerical examples, along with a reference to a simple counter-example scenario where the assumptions would fail. revision: yes

  2. Referee: [Theoretical results section] The sufficient conditions for practical asymptotic convergence are stated to be proved in the manuscript, yet the provided abstract and structure do not include the explicit assumption list, error-bound derivations, or the precise statement of the convergence theorem that would allow verification of whether the practical-stability gap is rigorously characterized.

    Authors: The sufficient conditions for practical asymptotic convergence are indeed proved in the manuscript. The full assumption list appears in Section II, the error bounds are derived in the proof of the main convergence result (Theorem 2 in Section IV, with supporting lemmas in the appendix), and the theorem explicitly characterizes the practical asymptotic convergence to the steady-state GNE while discussing the remaining gap to asymptotic stability. The abstract summarizes these contributions at a high level for brevity. We agree that greater explicitness in the abstract and main structure would aid verification. In the revision, we will expand the abstract to briefly reference the key assumptions and Theorem 2, and we will add a pointer in the introduction to the precise location of the error-bound derivations and the discussion of the practical-stability gap. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external turnpike results

full rationale

The central claims (recursive feasibility inferred from turnpike, sufficient conditions for practical asymptotic convergence) are derived from stated mild assumptions on dynamics and costs plus the turnpike phenomenon, which is invoked from prior literature rather than defined or fitted within the paper. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations that reduce the result to its own inputs appear in the provided structure or abstract. Numerical examples serve as validation, not as the source of the feasibility or stability claims. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the turnpike phenomenon holding under mild assumptions; no free parameters, new entities, or ad-hoc axioms beyond standard domain assumptions in receding-horizon control are visible from the abstract.

axioms (1)
  • domain assumption The turnpike phenomenon holds under mild assumptions on the system dynamics and cost functions
    Invoked to infer recursive feasibility without terminal ingredients.

pith-pipeline@v0.9.0 · 5400 in / 1175 out tokens · 71008 ms · 2026-05-13T03:26:17.251965+00:00 · methodology

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