Consider the following ODE: $\frac{d}{dt}\mathbf x = \mathbf {F(x,u)}$ in $t\in [0,t_{max}<\infty]$ with initial conditions $\mathbf x_0 = \mathbf x(0)$ and $\mathbf {F(x,u)}$ be smooth function of the vectors $\mathbf x$ and $\mathbf u$. Assume also that the matrix of the partial derivatives $\frac{\partial F_k}{\partial x_j}$ is negative definite within a sufficiently small neighborhood of the solution trajectories $\mathbf x =\mathbf {x(t,u|x_0)}$ of interest.
The set of controls are restricted to be at least piecewise smooth with finite number pieces. The set of singular points $\mathcal S (\mathbf u)$ are such that if $\mathbf {x \in \mathcal S(u)}$ then $\mathbf {F(x,u)}=0$.
My question is related to how close these singular points can be reached in finite time.
Specifically, starting from $\mathbf x_0$ and let the control $\mathbf u^1$ reach a singular point at $t=t^1$, that is $\mathbf x^1= \mathbf x(t^1) \in \mathcal S$. Consider all admissible controls whose singular points are reached at $t=t^1+t'$ and are in $\epsilon$ neighborhood of $\mathbf x^1$: $\mathbf {||x -x^1||} <\epsilon$ with $|t'|<\delta$.
(1) Can there be two controls (and trajectories) starting from the same $\mathbf x_0$ and reaching the same singularity at the same time ($\epsilon = $ and $\delta = 0$)?
(2) Is it true that if $\delta$ and $\epsilon$ are sufficiently small then there is no other neighboring trajectory starting from a different initial point than $\mathbf x_0$ still having its singularity within $[\epsilon,\delta]$?