Controllability Gramian

In control theory, we may need to find out whether or not a system such as

${\displaystyle {\begin{array}{c}{\dot {\boldsymbol {x}}}(t){\boldsymbol {=Ax}}(t)+{\boldsymbol {Bu}}(t)\\{\boldsymbol {y}}(t)={\boldsymbol {Cx}}(t)+{\boldsymbol {Du}}(t)\end{array}}}$

is controllable, where ${\displaystyle {\boldsymbol {A}}}$, ${\displaystyle {\boldsymbol {B}}}$, ${\displaystyle {\boldsymbol {C}}}$ and ${\displaystyle {\boldsymbol {D}}}$ are, respectively, ${\displaystyle n\times n}$, ${\displaystyle n\times p}$, ${\displaystyle q\times n}$ and ${\displaystyle q\times p}$ matrices for a system with ${\displaystyle p}$ inputs, ${\displaystyle n}$ state variables and ${\displaystyle q}$ outputs.

One of the many ways one can achieve such goal is by the use of the Controllability Gramian.