Nonlinear ODE with discontinuous dynamic

Let $G^+,G^-in mathcal{C}^1(mathbb{R}^n,mathbb{R}^n)$, and we define the following ODE : where $mathbb{R}^n=Acup Sigmacup B$, $A,B$ are two open sets and $Sigma$ is the hyperplane separating $A$ and $B$.
$$
begin{cases}
dot x(t)=G^-(x(t)) & text{if } x(t)in A\
dot x(t)=G^+(x(t)) & text{if } x(t)in B \
x(0)=x_0 in mathbb{R},
end{cases}
$$
And let $c_t$ is the crossing time : where $x(c_t)in Sigma$ and the velocity $dot x$ jumps to $B$.

I want to know if we can prove that $||dot x(c_t)||_{mathbb{R}^n}$ is bounded ?