# Finite time controllability Gramian

I have a question concerning the finite time controllability Gramian for linear continuous time-varying systems and it seems to me that maybe it is rather basic.

As known, the finite time controllability Gramian for the interval $$[t_0,t_1]$$ is defined as

$$\int_{t_0}^{t_1} \Phi(t_0,t)B(t)B(t)^T\Phi(t_0,t)^T \,\mathrm dt$$

and, as far as I understand, a system is controllable on $$[t_0,t_1]$$ if and only if the controllability Gramian is invertible. However, in Antsaklis & Michel's Linear System it is stated that a system is controllable at $$t_0$$ if and only if there exists $$t_1 > t_0$$ so that the controllability Gramian is nonsingular. And in the paper by Davis et al. 2009 on controllability, observability, realizability and stability of linear dynamical systems it is commented that a system that is controllable on $$[t_0,t_1]$$ may become uncontrollable if $$t_0$$ is decreased or $$t_1$$ is increased.

What I am getting confused about is the controllability Gramian a local or global statement?