= Controllability
{wiki}
<Controllability> asks whether the available <control inputs> can move a <dynamical system> between every pair of states in its declared <state space>. Fixing a duration $T>0$ gives the precise condition $\mathcal R_T(x_0)=X$ for every $x_0\in X$, where $X$ is the <state space> and $\mathcal R_T(x_0)$ its <reachable set>. With freely chosen duration, every pair must instead be connectable in some permitted finite time, which may depend on the pair. Always specify admissible <control inputs>, deadlines and any constraints along the path. https://fbswiki.org/wiki/index.php/State_Feedback[Åström and Murray, Feedback Systems: State Feedback, reachability definitions]{ref}
The surprising part is that one <control input> can control several state variables by acting at different times. For the <double integrator>, <acceleration> changes <velocity> directly, while the accumulated <velocity> changes <position>. Starting from rest, accelerate during the first half of the journey and brake during the second: the final <velocity> can vanish although the final <position> has changed. The figure is an analytic mathematical example in <dimensionless quantities>, with no experimental measurements.
\Image[/controllability-pulse-span.png]
{title=Timing one <control input> controls both <position> and <velocity>}
{description=The <double integrator> starts at $(p,v)=(0,0)$ and runs to $T=2$. Left: $u=1$ for one time unit, then $u=-1$ for one time unit. Middle: the analytic state path ends at $(1,0)$, after reaching $(1/2,1)$ when braking begins. Right: unit inputs on the first and second halves separately contribute $(3/2,1)$ and $(1/2,1)$ to the endpoint. Their <linear combinations> fill the plane when input amplitudes are unrestricted <real numbers>; subtracting the second contribution from the first gives $(1,0)$.}
<Controllability> is useful before trying to design a <controller>: a requested endpoint outside the <reachable set> cannot be attained by clever choice of <feedback control>. The <Kalman rank criterion> makes this an exact <linear algebra> calculation for the stated <linear time-invariant system> model. It concerns whole states, so attaining one <plant output> is not enough to establish <controllability>. In <quantum control>, the declared <state space> may be an orbit constrained by conservation laws; see <quantum controllability> and <unitary orbit of a density operator>.
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