I have repeatedly raised this, in my opinion, an extremely accurately formulated question here, but I have not received a qualified answer to it.
Let's take a simple gradient dynamical system:
$\frac{dx}{dt}=\frac{df}{dx}$
where $f=e^{-(x-x_*)^2}$ and $x_*$ - constant, that constant defining the position of the maximum.
The transition process in such a system is a transition from state $x(0)$ to state $x_*$.
We assume that we don't know the value $x_*$ in advance, as well as the function $f$ itself. How to make sure that transients in such a system always occur exponentially?
We can do anything: add control signals or add auxiliary variables.