Suppose we have $\dot x= F(t, u)x$ for $x\in\Bbb R^n$ and $F$ a continuous matrix in $t$ and $u$. In fact, in my case, there exists a smooth $f(t)$ s.t. $F(t, u)$ is invertible whenever $u \neq f(t)$. Is anything known about the controllability of this system near 0? In particular, when can we drive an initial $x_0\neq 0$ to some vector subspace $V\subset \Bbb R^n$ in finite time? In my case, $x_0$ can be scaled to any magnitude if necessary.
I found some papers online regarding something called the Sussman condition, which states that, if the system satisfies certain hypotheses, for any $t$ there exists a control $u$ which can drive the ball $B_\delta$ to $B_\epsilon$ in time $t$ for $\delta$ and $\epsilon$ small enough. This would more than suffice for my purposes, but there is no point in checking the conditions because a linear system cannot be driven to the origin with nice controls. Are there any known results on the matter? Could someone share some references?