Proof: is a real symmetric matrix. Consider the quadratic form
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
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
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.
The solution is
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
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
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
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 .
Now, use the Gramian
See Figure 1.1.
Figure 1.1: Two Possible Solutions for