An arXiv preprint develops a mathematical route for answering a basic control question: if a system can move in a few local directions, what larger set of motions can it reach? It connects the linear rank test to Lie brackets—algebraic operations on motion fields—and then to reachable-set statements for nonlinear systems on finite-dimensional matrix Lie groups.
The work is an arXiv preprint, version 1, dated 20 August 2026.
From rank to brackets
For the linear time-invariant (LTI) starting point, the paper states the Kalman-family condition as a rank test: the matrix built from B, AB, A²B and continuing through Aⁿ⁻¹B must have rank n. In ordinary terms, those successive directions must span the system’s full state space.
An oscillator example makes the test concrete. The displayed [B AB] matrix has rank two, and a suitable control over a finite interval can assign the oscillator’s position and velocity arbitrarily.
Once dynamics become nonlinear, repeated matrix products give way to Lie brackets. The paper organizes these bracket expressions in spaces of Lie monomials, tracking bounded occurrences of the control vector field, and reports the HLCC conditions as sufficient for small-time local controllability.
Rotations as a test case
Rotation groups supply the paper’s central finite-dimensional example. Skew-symmetric matrices are bracket-closed generators whose exponentials lie in SO(3); group commutators and the Baker-Campbell-Hausdorff (BCH) formula are used to represent bracket directions as finite group motions. That BCH step is local and subject to convergence.
The accompanying deterministic SO(3) simulation uses a nine-state system, the prescribed input u(t) = 0.2 cos t, and X(0) = I₃. The paper reports preserved orthogonality, X(t) ∈ SO(3), and each row tracing the unit sphere.
The calculation illustrates a geometric invariant, not a general reachability result. It uses one prescribed input and reports no reachability or performance comparison.
When local motion becomes a subgroup
For homogeneous right-invariant systems, concatenating controls gives the attainable set a semigroup structure—reachable motions can be combined in sequence. In the homogeneous case, time reversal and sign reversal supply inverses, so the attainable set is a subgroup.
When that attainable set is a subgroup, the paper identifies it with S, the subgroup associated with the Lie algebra generated by the system fields. It then invokes the Yamabe step: a subgroup connected by continuous paths, or arcwise connected, in a finite-dimensional Lie group is a Lie subgroup.
That leads to the stated if-and-only-if criterion for homogeneous right-invariant systems: the group must be connected, and the controlled fields must generate all of its Lie algebra. The paper does not present that criterion as a general rule for arbitrary nonlinear systems.
What the examples show
In SO(4), the localized direction B = C₀ begins in the (3, 4) plane. The displayed commutator sequence Cₖ, for k = 0,...,6, brings all six independent rotational planes of so(4) into the generated family. The example illustrates algebraic propagation, but it does not by itself replace a general controllability proof.
SO(7) extends the picture but stops short of a full result. B₇ begins in the final (6, 7) plane, and the displayed D₀ through D₃ sequence extends nonzero entries leftward along the chain. Full spanning of so(7) is posed as a controllability test rather than established.
The paper also checks its algebraic vocabulary on upper-triangular matrices. It concludes that b₃ is solvable and that the strictly upper-triangular algebra n₃ is nilpotent of class two. A further example shows that b₃ can be solvable but not nilpotent, with nested subspaces i₂ ⊂ i₁ ⊂ b₃ identified as ideals.
Where the analogy stops
The paper’s main boundary appears in general nonlinear systems. Their vector-field Lie algebras may be infinite-dimensional; the finite-dimensional analogy then lacks the Cayley-Hamilton termination used in the linear case and an unrestricted finite-dimensional Lie III correspondence. The paper identifies this as a structural limit and derives no general replacement theorem.
That boundary leaves several questions open: whether continued SO(7) commutators span all of so(7), what additional hypotheses could support comparable criteria for nonhomogeneous systems, and how finite-dimensional propagation arguments might be replaced or approximated for infinite-dimensional vector-field algebras. The BCH construction also remains local because of convergence.
Taken together, the preprint presents the Kalman family, Lie brackets, exponentials, BCH and generated subgroups as a common propagation architecture. Its examples make the bridge from local algebra to reachable sets concrete, while its caveats keep the conclusion tied to the finite-dimensional and homogeneous settings where the stated criteria apply.
Paper data and sources
Original title: Nonlinear Controllability and the Propagation of Local Information: From the Kalman Family to Lie Brackets, Rotation Groups, and Reachable Subgroups
Authors: Philippe Mullhaupt
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
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