= Double integrator
{title2=$\dot p=v,\quad\dot v=u$}
{wiki}
The <double integrator> is the <linear time-invariant system> in which one <control input> is integrated into <velocity>, and <velocity> into <position>. It is a mathematical idealization of one-dimensional motion under commanded <acceleration>; for a constant <mass> $m$ and net <force> $F$, <Newton's second law> gives $u=F/m$. The worked examples here use normalized <dimensionless quantities> and assume no <control disturbances>, rather than describing a laboratory record. https://www-personal.umich.edu/~dsbaero/others/25-DoubleIntegrator.pdf[Venkatesh G. Rao and Dennis S. Bernstein, Naive Control of the Double Integrator, pages 87–88]{ref}
Writing $x=(p,v)^T$ gives
$$
A=\begin{pmatrix}0&1\\0&0\end{pmatrix},\qquad
B=\begin{pmatrix}0\\1\end{pmatrix},\qquad
\mathcal C=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad
\det\mathcal C=-1.
$$
The <Kalman rank criterion> therefore gives <controllability> in every $T>0$ for unrestricted real <control inputs>. More concretely, choose constants $a,b$ on the first and second halves. Starting at $x_0=(p_0,v_0)^T$, direct <integration> gives
$$
\begin{pmatrix}p(T)-p_0-Tv_0\\v(T)-v_0\end{pmatrix}
=\begin{pmatrix}3T^2/8&T^2/8\\T/2&T/2\end{pmatrix}
\begin{pmatrix}a\\b\end{pmatrix}.
$$
The endpoint map has <determinant> $T^3/8\ne0$, so these two amplitudes can supply any desired endpoint displacement. For a rest-to-rest move of distance $d$, take $a=4d/T^2$, $b=-4d/T^2$. At $T=2$ and $d=1$ these are the two commands in the first figure. No impulse is needed; shorter deadlines require larger finite commands.
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