Consider the system : $\ddot{x}=u+\frac{x-x_1}{\|x-x_1\|^3}$ where $x \in \mathbb{R}^2$ and $x_1 \neq 0$ , The requirement is to make the origin globally asymptotically stable by choosing a continuous control signal $u$ so I can't just say let $u=-\frac{x-x_1}{\|x-x_1\|^3} + f(x,\dot{x})$ as it is not continuous near $x=x_1$.
I tried some possible choices of $u$ . However I wasn't able to get rid of the saddle points .
So my main question is : Can I use the fact that noises and disturbance rise in real world to say that these saddle points will be unstable when the controller is applied in real world ?
I would also appreciate it if someone pointed out a good choice of $u$ , but please notice that this is not my main objective .