= Reachability in control theory
{title2=$x_1\in\mathcal R(x_0)$}
<Reachability in control theory> asks about a particular initial state $x_0$ and target $x_1$: does some admissible <control input> make $x(T)=x_1$ at the declared time $T$, or at some allowed duration? <Controllability> requires this for every pair in the declared <state space>. This is a property of state evolution, distinct from graph <reachability>.
For a <linear time-invariant system>, solving the <ordinary differential equation> by a <matrix exponential> gives
$$
x(T)=e^{AT}x_0+\int_0^T e^{A(T-t)}B u(t)\,dt.
$$
The first term is the unforced endpoint. <Reachability in control theory> asks whether the <control input> can supply exactly the displacement $x_1-e^{AT}x_0$. The timing matters because the same command is carried by different amounts of subsequent dynamics. https://ee263.stanford.edu/archive/contr.pdf[Stephen Boyd, EE263: Controllability and state transfer, slide 18–3]{ref}
Here is a contrasting mathematical example with one shared <control input>:
$$
\dot z_1=u,\qquad\dot z_2=u,\qquad
A=\begin{pmatrix}0&0\\0&0\end{pmatrix},\qquad
B=\begin{pmatrix}1\\1\end{pmatrix},\qquad
\mathcal C=\begin{pmatrix}1&0\\1&0\end{pmatrix}.
$$
The <controllability matrix> has <rank of a matrix> one. Since $\frac{d}{dt}(z_1-z_2)=0$, the difference is invariant under every allowed <control input>. From $(0,0)$ the <reachable set> is the diagonal $z_1=z_2$ when input amplitudes are unrestricted: $(1,1)$ is reachable at $T=2$ using $u=1/2$, while $(1,0)$ is impossible at every duration. Having a <control input> appear in both state equations did not give independent control of both states. This <reachable set> is pictured alongside the bounded <double integrator> below.
Back to article page
