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?
The definition you present is clear and unambiguous. Notice that if the Gramian in positive-definite, then it is positive definite in any interval that contains the 1st one. It is thus straightforward to establish your statements concerning increasing and decreasing endpoints.
Alternative definitions of controllability can be made, but they cannot be more clear than the one you presented. I'd say forget them. It is only in the case of time-invariant systems that we can simply say "controllable" without mentioning the interval at all, in all other cases the definition that specifies both endpoints avoids needless complications.