For the dynamical system $\mathbf y' =\mathbf A(t,u)\mathbf y$, are there any known necessary conditions for controllability? I.e. what does $\mathbf A$ have to satisfy in order to ensure that, for any $y_0$, $y_1$, $\mathbf y(0)=y_0$ and for some time $T$ we have $\mathbf y(T)=y_1$. In this particular case, $\mathbf A$ can be assumed to be a polynomial in the scalar $u$ with time dependent matrix coefficients.
I have searched online but what I always run into is the Kalman rank conditions, which check for the controllability of the linearized system of an autonomous ODE. That is not the case here, given the dependence of $\mathbf A$ on time. To be fair, I do not require global controllability. If there are some linearization checks for non autonomous systems that allow me to control within a specific radius of a given trajectory would be enough. Essentially, I have a given ODE close to 0 and to infinity that I can modify on a compact set and I am trying to connect the given solutions at infinity and at the origin via this perturbation.