I am reading about control theory, more specifically, I am considering an affine drift-free control system of the form
$\dot{x} = \sum_{i=1}^{n} u_{i} A_{i}(x)$
Where $A_0,\dotsc,A_n$ are smooth vector fields on a manifold $M$ of dimension $n$. The system is locally controllable if the vector fields satisfy Hörmander's condition, that is to say, the span of the iterated Lie brackets should be of dimension $n$.
I think I fundamentally misunderstand something, so let us consider an example. Let us assume that we are on $\mathbb{R}^2$. Let us also assume that we are given two fields of the form $\Psi(x,y)\partial_x$ and $\Phi(x,y)\partial y$ where $\Psi$ and $\Phi$ are two functions that are smooth and nonvanishing. Here I don't need to iterate any Lie brackets, since the matrix $$ \begin{bmatrix} \Psi(x,y)& 0 \\ 0 &\Phi(x,y) \end{bmatrix} $$ since this matrix is of rank $2$, we have local controllability. Iterating Lie brackets is not going to give me anything new.
But this, to me, seems odd. How can this system be such that we can connect any points in $\mathbb{R}^2$? Does this not mean that control on manifolds are "done" and just reduces to finding vector fields? My concrete questions are:
Am I understanding and applying Hörmander's condition and Chow-Rashevskii in the correct way?
Am I misunderstanding how strong "local controllability" is?
You're looking at control systems with configuration space $M$ and control space $U$, defined by $$ \dot{x}(t)=f(x(t),u(t)) $$ Where $f:M\times U\to TM$ is a smooth function with $f(x,u)\in T_xM$.
Your example is one of the simplest possible cases: the function $f$ is an isomorphism, so the system is completely unconstrained, and any path can be obtained by suitable choice of the control parameter $u(t)$. As a result, it is locally controllable essentially because $M$ is locally path connected.
The "interesting" cases to which the Chow-Rashevskii theorem applies are when the image of $f$ is smaller: for instance if the vector fields $A_i$ do not span the tangent spaces, or if there are fewer than $n$ of them. In that case, the velocity at a point $x$ is constrained to a subspace of possible directions $\mathcal{D}_x\subset T_xM$, but it still may be possible to reach every point in a neighborhhod of $x$ through more complicated local motions.