= Controllability Gramian
{c}
{title2=$W_T=\int_0^T e^{As}BB^Te^{A^Ts}\,ds$}
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For $T>0$ and the constant <matrices> over the <real numbers> in the <Kalman rank criterion>, the finite-time <Controllability Gramian> accumulates the influence of all command times. Its <column space> equals that of the <controllability matrix>, and $W_T$ is a <symmetric positive-definite matrix> exactly when the <controllability matrix> has <rank of a matrix> $n$. If so, put $\delta=x_1-e^{AT}x_0$. Among <square-integrable functions> steering $x_0$ to $x_1$ at $T$, the command
$$
u_*(t)=B^Te^{A^T(T-t)}W_T^{-1}\delta
$$
minimizes $\int_0^T\|u(t)\|^2\,dt$, with value $\delta^TW_T^{-1}\delta$. Here $T$ in a matrix superscript denotes <transpose>. This cost is a squared input <norm>, not necessarily physical <energy>, and the optimizing command need not satisfy an amplitude bound. https://ee263.stanford.edu/archive/contr.pdf[Stephen Boyd, EE263: Controllability and state transfer, slides 18–29 to 18–30]{ref}
For the <double integrator>, $e^{As}B=(s,1)^T$, so direct <integration> gives
$$
W_T=\begin{pmatrix}T^3/3&T^2/2\\T^2/2&T\end{pmatrix},\qquad
\det W_T=\frac{T^4}{12}>0.
$$
For a rest-to-rest distance $d$, $u_*(t)=6d/T^2-12dt/T^3$ and the minimum squared input <norm> is $12d^2/T^3$. Compare the two constant commands in the <double integrator> example, whose cost is $16d^2/T^3$: both attain the target, but the smooth command costs less. At $T=2$, $d=1$, however, $u_*$ peaks at $3/2$, so it violates $|u|\le1$ while the two constant commands remain admissible. <Controllability>, <optimal control>, and constrained <reachability in control theory> answer different questions.
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