Vertical Control Systems on Tangent Bundles and Fiberwise Controllability

S.N. Stelmastchuk

arxiv: 2604.23414 · v1 · submitted 2026-04-25 · 🧮 math.OC

Vertical Control Systems on Tangent Bundles and Fiberwise Controllability

S.N. Stelmastchuk This is my paper

Pith reviewed 2026-05-08 07:33 UTC · model grok-4.3

classification 🧮 math.OC

keywords vertical control systemstangent bundlefiberwise controllabilitytransport operatorreachable setsrank conditionLie-algebraic criterionvector field lifts

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The pith

Vertical lifts of vector fields to tangent bundles reduce fiberwise controllability to a rank condition on the original fields.

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

The paper studies control systems on the tangent bundle generated by vertically lifting vector fields from a base manifold. These systems fix the base point and act only within each tangent space, reducing the problem to linear control on the fibers. Explicit solutions and reachable-set descriptions are derived, with fiberwise controllability shown to be equivalent to a rank condition on the base vector fields. When a complete drift is added to move the base point while vertical controls act on directions, a transport operator along the drift trajectory yields explicit solutions and a necessary-and-sufficient controllability criterion in terms of the transported fields, together with a Lie-algebraic sufficient test. A reader would care because the results separate positional motion from directional steering and supply direct, computable tests for when controls can reach any tangent vector.

Core claim

For vertical control systems on tangent bundles induced by lifts of vector fields, the vertical dynamics reduce to linear control problems on each tangent space, for which explicit solutions exist and reachable sets are fully characterized, making fiberwise controllability equivalent to a rank condition on the original vector fields. For lifted systems that combine a complete drift fixing the base trajectory with vertical controls, explicit solutions and reachable-set characterizations are obtained via a transport operator along the drift, producing a necessary and sufficient condition for fiberwise controllability in terms of transported vector fields along with a Lie-algebraic sufficient-c

What carries the argument

The transport operator along the drift trajectory, which carries vertical control vector fields forward while preserving the linear span needed to check rank conditions on the fibers.

Load-bearing premise

The vertical dynamics decouple into independent linear control problems on each tangent space and the transport operator along the drift trajectory is well-defined and structure-preserving.

What would settle it

A low-dimensional manifold and explicit vector fields where the rank condition holds at every point yet the reachable set in some fiber is a proper subspace, or where the transported fields span the tangent space but actual trajectories fail to reach every direction.

read the original abstract

We study control systems on the tangent bundle of a smooth manifold induced by vertical lifts of vector fields. The Vertical dynamics acts exclusively along the fibers, leaving the base point unchanged and reducing the system to a linear control problem on each tangent space, for which we obtain explicit solutions and characterize reachable sets, showing that fiberwise controllability is equivalent to a rank condition on the original vector fields. We then consider lifted systems combining complete drift and vertical controls, where the base trajectory is fixed by the drift and the control acts on tangent directions. For these systems, we derive explicit solutions and a complete characterization of reachable sets via a transport operator, yielding a necessary and sufficient condition for fiberwise controllability in terms of transported vector fields, together with a Lie-algebraic sufficient criterion.

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.

Desk Editor's Note private letter to a colleague

This paper gives explicit solutions and reachable-set characterizations for vertical lift systems on tangent bundles via a transport operator along the drift, but the core steps are standard geometric control.

read the letter

The main point is that vertical lifts of vector fields turn the control system into a linear one on each tangent space, and the paper works out explicit solutions plus reachable set descriptions for that. When you add a drift, they introduce a transport operator along the base flow to characterize when the system is fiberwise controllable. What stands out is the clean derivation of explicit solutions for the pure vertical case, where fiberwise controllability reduces to a rank condition on the original fields. For the drifted systems, the transport operator lets them move the control fields along the trajectory and give a necessary and sufficient rank condition on those transported fields, plus a Lie bracket based sufficient condition. This is presented directly without unnecessary abstraction. The work does well at making the reachable sets explicit rather than leaving them in terms of Lie algebras alone. The assumptions are standard: the drift is complete, and the transport is well-defined along its flow. The proofs seem to follow the usual lines from differential geometry and linear control theory, with no obvious gaps or circular steps. On the downside, the core techniques—vertical lifts and parallel transport—are familiar in geometric control, so the novelty lies in applying them specifically to fiberwise controllability and naming the transport operator for this purpose. Without concrete examples on particular manifolds or comparisons to existing results in the literature, it's hard to gauge how much this extends prior work. The paper stays tightly focused, which is good for clarity but means limited broader impact. This is aimed at researchers in mathematical control theory, particularly those interested in systems on tangent bundles or fiberwise properties. Someone working on controllability criteria for lifted systems would find the explicit conditions useful. It is solid enough on its own terms to deserve a serious referee, even if revisions might be needed for context or examples. I would recommend putting it through peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript studies control systems on the tangent bundle TM induced by vertical lifts of vector fields on a smooth manifold M. The vertical dynamics fix the base point and reduce to linear control problems on each tangent space, for which explicit solutions and reachable-set characterizations are derived, establishing that fiberwise controllability is equivalent to a rank condition on the original vector fields. For lifted systems combining a complete drift vector field with vertical controls, the base trajectory is fixed by the drift; explicit solutions and reachable-set characterizations are obtained via a transport operator along the drift flow, yielding a necessary and sufficient condition for fiberwise controllability in terms of the transported vector fields together with a Lie-algebraic sufficient criterion.

Significance. If the derivations hold, the paper supplies concrete geometric tools—explicit solutions, transport-operator characterizations, and rank/Lie-algebraic conditions—for analyzing fiberwise controllability on tangent bundles. These results strengthen the link between standard controllability rank conditions and lifted mechanical or velocity-level systems and could support further work on nonholonomic or Lagrangian control problems. The explicit constructions and complete characterizations are strengths that distinguish the contribution.

minor comments (3)

[Abstract] The abstract states that explicit solutions are obtained for the vertical dynamics and for the lifted systems, yet the form of these solutions is not indicated; a one-line expression or reference to the relevant theorem would improve immediate readability.
[Abstract] The Lie-algebraic sufficient criterion is announced but its precise statement (e.g., whether it involves the Lie algebra generated by the transported fields evaluated at a point) is not previewed; adding this detail in the abstract or introduction would help readers assess applicability.
[Introduction] No concrete low-dimensional example (e.g., on R^n or the circle) is mentioned in the abstract or introduction; including one would illustrate how the rank condition and transport operator operate in practice.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of the manuscript, the clear summary of its contributions, and the recommendation for minor revision. The assessment correctly identifies the core results on explicit solutions, reachable-set characterizations, and the rank/Lie-algebraic conditions for fiberwise controllability.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper's claims rest on standard geometric control constructions: vertical lifts of vector fields reduce the dynamics to linear control on each tangent fiber (with explicit solutions and rank conditions following directly from the definition of vertical lifts), and lifted systems use a transport operator along the fixed drift trajectory to characterize reachable sets via transported vector fields. These steps are derived from first-principles definitions of the tangent bundle, vertical lifts, and parallel transport without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. The necessary and sufficient fiberwise controllability condition is obtained by direct application of the transport operator to the original vector fields, which is independent of the target result. No ansatz is smuggled in, and the Lie-algebraic sufficient criterion is a standard consequence of the rank condition. The derivation chain is internally consistent and does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the work relies on standard assumptions from differential geometry and control theory with no apparent free parameters or invented entities.

axioms (1)

domain assumption The base manifold is smooth and the vector fields are smooth
Invoked as the setting for defining the tangent bundle, vertical lifts, and transport operators.

pith-pipeline@v0.9.0 · 5424 in / 1211 out tokens · 52685 ms · 2026-05-08T07:33:41.434546+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

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S. N. StelmastchukComplete Lifted Control Systems on Tangent Bundles: Non-Controllability and Chain Controllability. Submitted. 21

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A. A. Agrachev and Y. L. Sachkov,Control theory from the geometric view- point, Encyclopaedia of Mathematical Sciences Control Theory and Optimiza- tion, 87 II, Springer, Berlin, 2004; MR2062547

work page 2004

[2] [2]

Bullo and A

F. Bullo and A. D. Lewis,Geometric Control of Mechanical Systems: Mod- eling, Analysis, and Design for Simple Mechanical Control Systems, Springer Science+Business Media, New York, 2005

work page 2005

[3] [3]

Colonius and W

F. Colonius and W. Kliemann,The dynamics of control, Systems & Con- trol: Foundations & Applications, Birkh¨ auser Boston, Boston, MA, 2000; MR1752730

work page 2000

[4] [4]

Jurdjevic,Geometric control theory

V. Jurdjevic,Geometric control theory. Cambridge Studies in Advanced Mathematics, 52. Cambridge University Press, Cambridge, 1997. xviii+492 pp

work page 1997

[5] [5]

Yano and S

K. Yano and S. Ishihara,Tangent and cotangent bundles: differential geome- try. Pure and Applied Mathematics, No. 16. Marcel Dekker, Inc., New York,

work page

[6] [6]

D., Sontag,Mathematical control theory, second edition, Texts in Applied Mathematics, 6, Springer, New York, 1998; MR1640001

E. D., Sontag,Mathematical control theory, second edition, Texts in Applied Mathematics, 6, Springer, New York, 1998; MR1640001

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S. N. StelmastchukComplete Lifted Control Systems on Tangent Bundles: Non-Controllability and Chain Controllability. Submitted. 21

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