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On the one hand, the question is simple, on the other hand, I need the help of specialists in control theory.

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.

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  • $\begingroup$ Define exponentially. Do you want to reach the steady-state faster? is the transient allowed to overshoot? Can we assume $f$ is an exponential equation? The simplest way to increase rising time is to just put a gain in the system: $\frac{dx}{dt} = \alpha\frac{df}{dx}$, since the derivative should be zero if the state has reached the unknown steady state $\endgroup$
    – Petrus1904
    Commented Jan 8, 2021 at 16:49
  • $\begingroup$ @Petrus1904 https://www.wolframalpha.com/input/?i=x'%3DD(exp(-(x-1)^2)%2Cx)%2Cx(0)%3D-1 Take a look at this example. I want the transition from any initial state to the final state to occur exponentially, i.e.: $x(t)=(x(0)-x_*) \cdot exp(-t) + x_*$ But keep in mind, $x_*$ it is not known in advance, the gradient system is looking for this point. $\endgroup$
    – ayr
    Commented Jan 8, 2021 at 16:53

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