A useful way to compare linear and nonlinear controllability is to ask how much information near a point can be propagated into a statement about finite-time reachability. In a linear time-invariant system, a finite family of directions — $B,AB,\ldots,A^{n-1}B$ — completely determines controllability. The deeper question is whether an analogous finite collection of local directions can determine reachable sets in a nonlinear system.

The natural nonlinear replacement of the Kalman family is the family of iterated Lie brackets of the vector fields defining the system. This replacement is exact in the linear case, but nonlinear vector fields introduce two essential difficulties: there is no nonlinear analogue of Cayley–Hamilton that forces the bracket family to terminate, and exact trajectories involve increasingly complicated iterated integral structures. The algebra generated by the vector fields may therefore be infinite dimensional.

Rotation matrices isolate the finite-dimensional mechanism in a setting where every object can be calculated explicitly. The Lie algebra $\mathfrak{so}(n)$ consists of skew-symmetric matrices, the corresponding Lie group is $\operatorname{SO}(n)$, the exponential map is the matrix exponential, and commutators of skew-symmetric matrices are again skew symmetric. This makes the rotation group an ideal laboratory for the slogan “local information propagates.” The examples below are therefore treated as part of the main argument rather than as secondary illustrations.

Consider the single-input LTI system

Let $\Phi(x_{0},t,u)$ denote the solution generated by an input $u(\cdot)$. The unrestricted-time reachable set from the origin is

and the reachable set at a specified time is

Small-time local controllability at the origin means that $\mathcal{R}(0,t)$ contains a neighborhood of the origin for every $t>0$. Global controllability means that every target state can be reached in some finite time.

Let $M$ be a smooth manifold. A vector field $f$ assigns a tangent vector $f(p)\in T_{p}M$ at each point. In coordinates, we use the Lie bracket

It is antisymmetric and satisfies Jacobi’s identity, so the vector fields form a Lie algebra.

The linear case is recovered immediately. For $f(x)=Ax$ and the constant vector field $g(x)=B$,

Hence the Kalman family is exactly an iterated-bracket family, up to alternating signs.

For a scalar-input system

we introduce spaces $S_{k}(f,g)$ consisting of linear combinations of Lie monomials in $f$ and $g$ containing $g$ at most $k$ times. These spaces organize the bracket information needed in the local controllability condition.

The following conditions are useful in the scalar-input setting, called the Hermes Local Controllability Conditions (HLCC):

1. (HLCC 1)

   $x_{0}$ is a regular equilibrium for some admissible constant control $\bar{u}$, so $f_{0}(x_{0})+\bar{u}f_{1}(x_{0})=0$;

2. (HLCC 2)

   $\dim\operatorname{Lie}(f_{0},f_{1})(x_{0})=\dim M$;

3. (HLCC 3)

   for the increasing sequence $S_{k}(f_{0}+\bar{u}f_{1},f_{1})(x_{0})$, equality $S_{k}=S_{k+1}$ is required whenever $k$ is odd.

Sussmann’s 1983 theorem states that these conditions imply small-time local controllability [10]. The theorem is deliberately used as motivation rather than proved: its proof is much more elaborate than the linear rank test and exposes precisely the limitations of ordinary finite-dimensional Lie theory. Sussmann’s later general theorem develops the nilpotent-approximation viewpoint further [11].

The rotation-matrix sequence is not merely an illustration. It is the concrete finite-dimensional model through which the correspondence between a Lie algebra, its exponential map, and a Lie group becomes visible.

Finite-dimensional Lie theory supplies a connected simply connected Lie group for every finite-dimensional Lie algebra. In the matrix setting the correspondence is concrete: matrix commutators live in the Lie algebra and the matrix exponential maps toward the group.

This mechanism cannot simply be transferred to the full Lie algebra of smooth vector fields. The algebra of vector fields is typically infinite dimensional. The cited example of Sergeraert [9] shows that even diffeomorphisms infinitely tangent to the identity need not embed into a one-parameter group. Thus the heuristic “vector field $\leftrightarrow$ one-parameter subgroup” has genuine limitations outside the finite-dimensional setting.

A direct elementary warning is provided by the vector fields With

repeated brackets produce terms of successively higher degree, including $x_{1}^{5},x_{1}^{6},x_{1}^{7}$, showing concretely how an infinite family of independent vector fields can arise.

A central example collects a nine-state nonlinear system by collecting the states into a $3\times 3$ matrix

With the skew-symmetric matrices

the system is written

This matrix representation exposes the invariant rotation-group structure.

Since $T_{1}+u(t)T_{2}$ is skew symmetric for every time, the flow preserves orthogonality. Indeed,

With $X(0)=I$, one therefore has $X(t)\in\operatorname{SO}(3)$ throughout the simulation. This identity explains the geometry seen in the three Mathematica drawings: each row of $X(t)$ remains a unit vector and hence traces a curve on the unit sphere, while the three rows remain mutually orthogonal.

The corresponding Mathematica construction is included because it makes the simulation reproducible:

Repeating the final command for rows two and three produces figure 1.

Consider the right-invariant control system

on a finite-dimensional Lie group $G$. Let $e$ be the identity, $\mathcal{R}(e)$ the attainable set from $e$, $\mathfrak{s}$ the Lie subalgebra generated by $X_{0},\ldots,X_{m}$, and $S$ the connected subgroup associated with $\mathfrak{s}$.

Controllability reduces to two structural questions: when is $\mathcal{R}(e)$ a subgroup, and, once it is a subgroup, when is it the whole of $G$?

The central finite-dimensional statement is the following: if $\mathcal{R}(e)$ is a subgroup, then it coincides with the subgroup $S$ associated with the Lie algebra generated by the system fields. This is the crucial step in translating local algebraic generation into global reachability.

The subtle point is that an abstract subgroup of a Lie group need not automatically inherit the required embedded Lie-subgroup structure. Yamabe’s theorem gives the needed bridge: an arcwise connected subgroup of a Lie group is a Lie subgroup [13]. Since controlled trajectories provide arcs from the identity, $\mathcal{R}(e)$ is arcwise connected.

Let $A=\mathcal{R}(e)$ and let $\mathfrak{a}$ denote its Lie algebra. The inclusion $A\subseteq S$ gives $\mathfrak{a}\subseteq\mathfrak{s}$. Conversely, take constant controls with components $a_{i}\in\{\pm 1\}$ and define

The positive-time curve $\exp(tX(a))$ lies in $A$. If $A$ is a group, negative times lie there as well, so $X(a)\in\mathfrak{a}$. The resulting family of $X(a)$ generates the original system directions, hence $\mathfrak{s}\subseteq\mathfrak{a}$. Therefore

This is the classical Lie-group control mechanism developed by Jurdjevic and Sussmann [12].

Yamabe’s theorem states that an arcwise connected subgroup $A$ of a finite-dimensional Lie group $G$ is a Lie subgroup [13]. For controllability this result is decisive: an attainable set may first be known only as a subgroup and as a union of controlled arcs, while the Lie-algebra argument requires a genuine Lie subgroup. The following proof sequence makes the finite-dimensional mechanism explicit.

For a homogeneous right-invariant system, $\mathcal{R}(e)$ is already a subgroup. Combining this with equation 15 gives the criterion:

A homogeneous right-invariant system is controllable on $G$ if and only if $G$ is connected and the Lie algebra generated by the controlled vector fields is all of $\mathfrak{g}$.

This is the closest finite-dimensional nonlinear analogue of the Kalman rank test in this setting.

Consider

with

Both are in $\mathfrak{so}(4)$. Set

The first seven members are

The algebraic pattern is clearer in graphical form. In figure 2, each cell shows a matrix entry; white cells are zero, while the signed nonzero entries are colored according to magnitude. The initial control direction $B=C_{0}$ is supported only on the $(3,4)$ coordinate plane. The first commutator moves this support to the $(2,4)$ plane, and subsequent commutators populate further coordinate pairs until all six independent rotational planes of $\mathfrak{so}(4)$ appear across the generated family.

This is the rotation-group analogue of the linear sequence $B,AB,A^{2}B,\ldots$. The relevant linear operator on the Lie algebra is now

The family generated from $B$ by repeated application of $\operatorname{ad}_{A}$ is therefore the natural finite-dimensional bracket counterpart of the Kalman family. In particular, the graphical sequence shows not merely growth of numerical coefficients but propagation of support through distinct coordinate planes.

The same construction extends naturally to $\operatorname{SO}(7)$. Let $A_{7}$ be the skew-symmetric nearest-neighbor chain

and localize the control generator in the final $(6,7)$ plane,

Define $D_{0}=B_{7}$ and $D_{k+1}=[A_{7},D_{k}]$. The first four members are

Thus the bracket depth has a direct spatial interpretation along the chain: $D_{0}$ acts only at the end, $D_{1}$ reaches one coordinate farther, $D_{2}$ reaches the fourth coordinate, and $D_{3}$ reaches the third. The full matrix values are shown graphically in figure 3.

This example makes “propagating local information” literal. Locality refers to the support of $B_{7}$ in a single rotational plane; propagation refers to the successive appearance of nonzero skew-symmetric entries in coordinate planes farther from that end. The same mechanism can be continued to higher bracket depth, and in a controllability test one asks whether the Lie algebra generated by the available directions spans all of $\mathfrak{so}(7)$.

For compact Lie groups, compactness can restore group-like reachability properties even when a drift $X_{0}$ is present. In particular, in the compact Lie-group case one obtains a result of the same flavor as in the homogeneous case, whereas for a general nonhomogeneous system additional hypotheses are needed.

It is useful to distinguish three levels: finite-dimensional homogeneous systems, finite-dimensional nonhomogeneous/decomposable systems, and general infinite-dimensional vector-field algebras. The first level is completely controlled by the generated Lie algebra; the later levels require progressively more refined tools.

The linear and finite-dimensional Lie-group theories share a common architecture. In the linear case, Cayley–Hamilton ensures that the directions generated by repeated action of $A$ form a finite family. On $\operatorname{SO}(n)$ and other finite-dimensional Lie groups, closure under Lie brackets and the Lie correspondence play the analogous role. BCH and group commutators explain how directions in the Lie algebra are converted into finite motions in the group.

The $\operatorname{SO}(3)$ simulation makes this correspondence geometric: skew-symmetric instantaneous generators preserve the orthogonal constraint, so the trajectory remains on the group and each matrix row remains on the sphere. The $\operatorname{SO}(4)$ and $\operatorname{SO}(7)$ examples make the algebraic propagation visible: a localized rotational input direction spreads through iterated commutators.

For a general nonlinear system, however, the Lie algebra of vector fields may be infinite dimensional. There is then no Cayley–Hamilton theorem to terminate the bracket generation and no unrestricted finite-dimensional Lie III correspondence to turn the algebra into a globally manageable group. This is precisely the point at which Sussmann’s formal Lie series, filtrations, nilpotent approximations, and symmetry arguments become necessary [10, 11].

The results above answer a common question at several levels: how can information known infinitesimally be propagated into a statement about finite-time reachability?

For LTI systems, the answer is the Kalman family and Cayley–Hamilton. For finite-dimensional matrix Lie groups, the answer is the Lie algebra, the exponential map, group commutators, BCH, and the subgroup generated by the controlled directions. The rotation examples are the most concrete expression of this mechanism: skew-symmetric matrices generate rotations; their commutators generate new infinitesimal rotations; periodic controlled dynamics remain on $\operatorname{SO}(3)$; and repeated adjoint operations on $\operatorname{SO}(4)$ and $\operatorname{SO}(7)$ visibly propagate a localized actuation direction through the algebra.

The analogy breaks when the generated vector-field algebra is infinite dimensional. The resulting failure is not incidental but structural, and it explains the technical depth of general nonlinear local controllability theorems. The finite-dimensional rotation examples therefore serve a double role: they provide exact controllability results in their own right and expose, in the cleanest possible setting, the mechanism that more general nonlinear theory attempts to recover by approximation.