= Kalman rank criterion
{c}
{title2=$\operatorname{rank}\mathcal C=n$}
For a finite-dimensional continuous-time <linear time-invariant system> $\dot x=Ax+Bu$ on $\mathbb R^n$, with constant <matrices> $A,B$ over the <real numbers>, no state constraints, and every <piecewise continuous function> $u:[0,T]\to\mathbb R^m$ allowed as a <control input> without an amplitude bound, <controllability> in any fixed $T>0$ is equivalent to the <controllability matrix> having <rank of a matrix> $n$. The same equivalence holds with all <square-integrable functions> as inputs. A stronger useful statement is
$$
\mathcal R_T(x_0)=e^{AT}x_0+\operatorname{im}\mathcal C,\qquad T>0,
$$
where $e^{AT}$ is a <matrix exponential> and $\operatorname{im}\mathcal C$ is the <column space>. Thus a <rank of a matrix> $r<n$ leaves only an $r$-dimensional <affine subspace> reachable at that time. https://ocw.mit.edu/courses/16-30-feedback-control-systems-fall-2010/4bebd71d575bdde1384713e9efb400cc_MIT16_30F10_lec10.pdf[MIT 16.30/31, Feedback Control Systems: Controllability, pages 10–1 to 10–2]{ref}
The <Kalman rank criterion> allows arbitrarily large commands as $T$ shrinks. It does not assert <controllability> with bounded <control inputs>, nor test a general nonlinear <dynamical system> just by counting its inputs. For the <double integrator>, $B=(0,1)^T$ and $AB=(1,0)^T$ give two independent directions, despite there being only one <control input>.
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