Assume two parallel systems of the form $\dot{x}=f(x_t)+u_t$ and $\dot{y}=f(y_t)+u_t$. The input that is added to these two systems is the same. Also assume that $f$ is globally unstable, meaning starting at any initial condition $x_0$ except the equilibrium point ($x=0$), the zero input system' states; i.e., $\dot{x}_t=f(x_t)$ diverges to infinity. Does there exist any $u_t$ that can stabilize both parallel systems with different initial conditions $x_0\neq y_0$?
For linear systems obviously the answer is no because by subtracting both equations we get $\dot{y}-\dot{x}=A(y_t-x_t)$ and with unstable $A$ and $y_0-x_0\neq 0$ that is not orthogonal to unstable eigenvector, the difference diverges to infinity. Therefore, at least one of the systems will diverge to infinity. What can we say for nonlinear systems?