Proof: 
is a real symmetric 
matrix. Consider
the quadratic form
where
We have
Function 
is the same as the one appearing in the proof of
theorem (1.3.1). Hence, by equation (1.8)
of the above proof, the statement
holds if and only if the system is controllable. We thus have that, whenever controllability holds then
and
That is is a positive-definite sysmmetric matrix. Then all of its
eigenvalues are positive real, hence is invertible and its
rank is 
.
Replace 
by the variable 
, and consider
The right hand side is differentiable with respect to , yielding,
by Leibniz formula
or
The above is a matrix differential equation which can be used to
compute 
by one of the numerical methods for differential
equations. In some simple cases can be computed numerically.
Example 
.
Given
let
then
or
The solution is
and
So the system is controllable. In fact, this can be verified much faster
by computing 
.
By a change of variables 
, 
, the formula
for becomes
So in our example we have
so
which coincides with the previous solution.
Note: For any 
The integrand has a nonnegative value for all 
. Hence, 
is monotone nondecreasing with . Therefore, for any 
,
is either growing with or is constant at
some subintervals of the real line.
When the matrix 
has all of its eigenvalues in the left half plane,
then 
as 
, so there are
constants 
such that
Then
Now
For any , the function is monotone
nondecreasing and bounded for all . By a known theorem
in calculus, for every the function
converges to a limit. Because is arbitrary, this implies
can be obtained as a steady-state solution of the differential
equation (1.12), by setting 
. We then
have that satisfies a matrix Lyapunov equation
The controllability Gramian is useful in many instances. Consider,
for example, the problem of finding control that
steers a systems from the initial state 
to
a final state 
. Such a control must satisfy
The solution of this problem is not unique. We can assume the form of 
leaving some parameter free, and then determine these parameter from
equation (1.13) above. One way of doing this is to assume
to be a piece-wise constant function on subintervals of
, and determin the amplitudes of on these subintervals from
condition (1.13), see the example below. Another way is to
assume that has the form
Substituting (1.14) to (1.13) we have
and
so
Then control has the following interpretation: the control is
proportional to the difference between the desired state 
and
the predicted endpoint of a free trajectory starting from 
.
Example 
.
Let
First assume
or
Hence
Now, use the Gramian
See Figure 1.1.
Figure 1.1: Two Possible Solutions for
