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arxiv: 1907.01786 · v1 · pith:EK7DEWN5new · submitted 2019-07-03 · 🧮 math.OC

Towards Velocity Turnpikes in Optimal Control of Mechanical Systems

Timm Faulwasser , Kathrin Fla{\ss}kamp , Sina Ober-Bl\"obaum , Karl Worthmann This is my paper

Pith reviewed 2026-05-25 10:36 UTC · model grok-4.3

classification 🧮 math.OC
keywords optimal controlmechanical systemsturnpike propertytrim solutionsvelocity steady stateshyperbolic turnpikefinite horizon optimization
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The pith

For any finite horizon, optimal solutions in a mechanical system example correspond to optimal trim solutions that form velocity turnpikes.

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

The paper introduces the ideas of velocity steady states, treated as partial steady states, and hyperbolic velocity turnpike properties to study optimal control in mechanical systems. It shows in one concrete example that both the optimal solution and the turnpike orbit match optimal trim solutions no matter how long or short the time horizon is. This is presented as the first combination of trim primitives with time-varying turnpike analysis. Readers might care because it points to a structure that could simplify solving optimal control problems by reducing them to these velocity-based trim orbits rather than full equilibria.

Core claim

We show for a specific example that, for all finite horizons, both the (essential part of the) optimal solution and the orbit of the time-varying turnpike correspond to (optimal) trim solutions. The concepts of velocity steady states and hyperbolic velocity turnpike properties are proposed for analysis and control of mechanical systems.

What carries the argument

Velocity steady states as partial steady states where only velocities are at equilibrium, combined with the hyperbolic velocity turnpike property that describes attraction to time-varying turnpikes.

If this is right

  • For all finite horizons the essential optimal solution is a trim solution.
  • The orbit of the time-varying turnpike is an optimal trim solution.
  • This correspondence holds in the chosen mechanical system example.
  • Trim primitives can be combined with time-varying turnpike properties for control design.

Where Pith is reading between the lines

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

  • If the property generalizes, it may allow reducing infinite-horizon problems to finding appropriate trim solutions.
  • Similar velocity turnpikes could appear in other underactuated mechanical systems with similar dynamics.
  • Computation of optimal controls might be simplified by precomputing families of trim solutions instead of solving full boundary value problems.

Load-bearing premise

The correspondence between optimal solutions, turnpikes, and trim solutions observed in the example will hold under the system dynamics and cost without needing extra unstated conditions.

What would settle it

A numerical computation for the example system at some finite horizon T where the optimal trajectory does not match a trim solution.

Figures

Figures reproduced from arXiv: 1907.01786 by Karl Worthmann, Kathrin Fla{\ss}kamp, Sina Ober-Bl\"obaum, Timm Faulwasser.

Figure 1
Figure 1. Figure 1: Numerical solution of the illustrative example for T = 20. and, thus, v ? (t) = (cosh(t) − 1)λ1(0) − sinh(t)λ2(0) =  sinh(t) + sinh(T − t) − sinh(T) 2(cosh(T) − 1) − T sinh(T)  q. ˜ A direct calculation yields the first two derivatives of v ? (t): v ?0 (t) =  cosh(t) − cosh(T − t) 2(cosh(T) − 1) − T sinh(T)  q, ˜ v ?00(t) = −  sinh(T − t) + sinh(t) 2(cosh(T) − 1) − T sinh(T)  q. ˜ For T > 0, the deno… view at source ↗
Figure 2
Figure 2. Figure 2: Numerical solution of the illustrative example for T ∈ {5, 10, 20, 40, 80}. analogous argumentation). Next, we show |v ? (T /2)| ≤ Cq˜/T ∀ T > 0 with Cq˜ = 3|q˜|/2: T|v ? (T /2)| |q˜| = T  2 sinh(T /2)(cosh(T /2) − 1) T sinh(T) − 2(cosh(T) − 1) = T(sinh(T) − 2 sinh(T /2)) (T − 2) sinh(T) + 2(1 − e−T ) = P∞ k=2 T 2k (2k−1)!  1 − 1 2 2(k−1)  P∞ k=2 T2k (2k−1)! [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: We consider q0 = 0 and qT = 5 as boundary conditions on the configuration and a fixed final time T = 20. If the boundary velocities are chosen to exactly match 11 [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Hovercraft parallel parking example. [2] D. Carlson, A. Haurie, and A. Leizarowitz. Infinite Horizon Optimal Control: De￾terministic and Stochastic Systems. Springer, 1991. [3] T. Damm, L. Gr¨une, M. Stieler, and K. Worthmann. An exponential turnpike theorem for dissipative optimal control problems. SIAM Journal on Control and Optimization, 52(3):1935–1957, 2014. [4] R. Dorfman, P. Samuelson, and R. Solow.… view at source ↗
read the original abstract

The paper proposes first steps towards the formalization and characterization of time-varying turnpikes in optimal control of mechanical systems. We propose the concepts of velocity steady states, which can be considered as partial steady states, and hyperbolic velocity turnpike properties for analysis and control. We show for a specific example that, for all finite horizons, both the (essential part of the) optimal solution and the orbit of the time-varying turnpike correspond to (optimal) trim solutions. Hereby, the present paper appears to be the first to combine the concepts of trim primitives and time-varying turnpike properties.

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 / 2 minor

Summary. The manuscript introduces the notions of velocity steady states (as partial steady states) and hyperbolic velocity turnpike properties for optimal control problems on mechanical systems. It then demonstrates, for one concrete mechanical example and every finite horizon, that the essential part of the optimal trajectory and the orbit of the associated time-varying turnpike both coincide with optimal trim solutions. The work positions itself as the first to combine trim primitives with time-varying turnpike analysis.

Significance. If the numerical or analytic verification for the chosen example is correct, the paper supplies a concrete, reproducible illustration that velocity turnpikes can be realized by trim primitives. This supplies a useful benchmark case for subsequent theoretical work on time-varying turnpikes, even though no general theorem is claimed.

minor comments (2)
  1. [Example section (around the trim-solution verification)] The abstract states that the correspondence holds 'for all finite horizons' yet the manuscript should explicitly state the range of horizons that were actually computed and whether the hyperbolic property was checked analytically or only numerically for the example system.
  2. [Preliminaries / Definitions] Notation for the velocity steady state and the time-varying turnpike orbit should be introduced with a short table or diagram that distinguishes the full state, the velocity component, and the trim primitive.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, as well as for the recommendation of minor revision. The assessment that the work provides a useful benchmark case is appreciated.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central claim is explicitly restricted to demonstrating a correspondence between optimal solutions and trim solutions for one specific mechanical system example across finite horizons. No general theorem is asserted, no parameters are fitted and then relabeled as predictions, and no load-bearing step reduces to self-definition, self-citation chains, or imported uniqueness results. The work is framed as proposing concepts and providing an illustrative case, rendering the derivation self-contained without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review reveals no explicit free parameters, axioms, or invented entities; the new concepts (velocity steady states, hyperbolic velocity turnpike) are definitional rather than derived from prior fitted quantities.

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Reference graph

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