I have a state-affine, time-independent system of the form $$ \mathbf x'(t) = \mathbf F(u) \mathbf x(t)$$ where $\mathbf F$ is a matrix polynomial in the scalar $u$. Now, suppose that $u$ is such that $\mathbf x(t_0) =\mathbf x_0$ and $\mathbf x(t_1) = \mathbf x_1$. Under what conditions on a new matrix $\mathbf P(t)$ does there exist a new control $u^*$ such that $(\mathbf x(t_0), \mathbf x(t_1)) = (\mathbf x_0, \mathbf x_1)$ with $\mathbf x$ satisfying $$\mathbf x'(t) = (\mathbf F(u^*)+\mathbf P(t)) \mathbf x(t)\ ?$$ On Google I have only found papers on the preservation of controllability after a perturbation that vanishes faster than $\mathbf x$, which is not the case here. Furthermore, I really do not care about preserving controllability, my main concern is knowing whether or not the particular states $(\mathbf x_0, \mathbf x_1)$ remain connected by some control.
The reason I believe this should hold in my case is that $\mathbf P$ can have arbitrarily small norm if $t_0$ is large enough, so its effect on the system should be small enough for a fixed-point theorem to finish the job. Unfortunately, I have no idea which operator to pick to try and find a fixed point. One naive try would be to linearize around the trajectory generated by $\mathbf x'(t) = (\mathbf F(u)+\mathbf P(t)) \mathbf x(t)$ and show that the resulting linear system is controllable in a tubular neighborhood of $\mathbf x(t)$. If this neighborhood is large enough to include $\mathbf x_1$ I would be done. Yet, to estimate this size, I need bounds on the control Gramian of the linearized system, and getting them seems intractable.
Is there any way to show this or are there any known results on the subject?
EDIT I forgot to mention that $u$ can vary in time over $[t_0, t_1]$ and therefore so can $u^*$. However, any ideas regarding $u$ constant (but $u^*$ either constant or time varying, whichever is easier) would be appreciated.