Show that two definitions of the controllability Gramians are equivalent

Question

Given a dynamical system $\dot x = Ax + Bu$, I've noticed that several authors define the controllability Gramian as:

$$W = \int_0^te^{A(t-\tau)} BB^T e^{A^T(t-\tau)}d\tau$$

while others define it as, $$W = \int_0^te^{A\tau} BB^T e^{A^T\tau}d\tau$$

More strangely, I see other places define them simultaneously as follows: $$W = \int_0^te^{A(t-\tau)} BB^T e^{A^T(t-\tau)}d\tau = \int_0^te^{A\tau} BB^T e^{A^T\tau}d\tau$$

For instance: https://en.wikipedia.org/wiki/Controllability_Gramian

What gives? I know that this integral is not easy to evaluate. So how do I show that they are actually equivalent?

Answer 1

One integral is obtained from another using a change of variables: $$ \int_0^te^{A(t-\tau)} BB^T e^{A^T(t-\tau)}d\tau=\left| s=t-\tau\atop ds=-d\tau \right|= -\int_{t-0}^{t-t}e^{As} BB^T e^{A^Ts}ds $$ $$ =-\int_{t}^{0}e^{As} BB^T e^{A^Ts}ds=\int^{t}_{0}e^{As} BB^T e^{A^Ts}ds. $$