= Reachable set
{title2=$\mathcal R_T(x_0)=\{x(T;x_0,u):u\text{ admissible}\}$}
The <reachable set> $\mathcal R_T(x_0)$ collects all endpoints attainable from $x_0$ at exactly $T$ by admissible <control inputs>. If arrival any time up to $T$ is allowed, use $\mathcal R_{\le T}(x_0)=\bigcup_{0\le t\le T}\mathcal R_t(x_0)$, with $\mathcal R_0(x_0)=\{x_0\}$. These definitions differ: a general <dynamical system> may drift away after early arrival, so waiting need not preserve the target. The <reachable set> is always relative to a declared input class and any constraints on the entire path.
For the <double integrator> starting from $(0,0)$, now require <measurable functions> as <control inputs> with $|u(t)|\le1$ <almost everywhere>. Write $(p,v)$ for its endpoint at $T>0$. Two <integrals> determine the endpoint:
$$
v=\int_0^T u(t)\,dt,\qquad p=\int_0^T(T-t)u(t)\,dt.
$$
Our mathematical example has the exact <reachable set>
$$
\mathcal R_T(0)=\left\{(p,v): |v|\le T,\quad
-\frac{T^2}{4}+\frac{Tv}{2}+\frac{v^2}{4}\le p\le
\frac{T^2}{4}+\frac{Tv}{2}-\frac{v^2}{4}\right\}.
$$
To see the upper boundary, fix $v$: put $u=1$ first and $u=-1$ last, switching at $s=(T+v)/2$. This maximizes the weighted <integral> for <position> because early commands have the largest weight $T-t$; the required net <velocity> fixes $s$. Formally, $(T-t)-(T-s)=s-t$ has the same sign as this maximizing command, so changing to any other admissible command with the same <integral> cannot increase $p$. Reversing the order gives the lower boundary. Pointwise <convex combinations> of these two commands preserve $v$ and attain every intermediate $p$, so the formula describes the whole region, not merely a bound. The extreme commands are <bang-bang control>.
\Image[/reachability-bounded-and-shared.png]
{title=An amplitude bound limits endpoints; a shared input can remove a state direction entirely}
{description=Analytic mathematical examples, not measured data. Left: the exact <double integrator> <reachable set> from rest at $T=2$ with $|u|\le1$. Each curved boundary comes from one switch between $+1$ and $-1$. At final <velocity> $v=0$, the available <position> interval is $[-1,1]$, excluding $(3/2,0)$. Right: $\dot z_1=\dot z_2=u$ only reaches the diagonal from zero. Unrestricted inputs fill the entire line; the same bound $|u|\le1$ at $T=2$ leaves only its segment from $(-2,-2)$ to $(2,2)$.}
In particular, the bounded <double integrator> can move from rest to <position> $d$ and rest again exactly when $|d|\le T^2/4$. Its minimum duration is therefore $2\sqrt{|d|}$ in these normalized units. With freely chosen time every rest-to-rest distance is attainable; with a fixed deadline many are excluded. Changing the amplitude bound to $|u|\le a$ with $a>0$ scales this condition to $|d|\le aT^2/4$. This shows why specifying the controls matters even when the <Kalman rank criterion> gives full <rank of a matrix>.
In closed <quantum control>, a <density operator> <reachable set> lies inside its <unitary orbit of a density operator>, since coherent evolution preserves the <spectrum>. That orbit need not be fully reachable under the available <control inputs>.
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