# Controlling the dynamics of nonlinear systems with an unknown steady state

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?

• 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 Commented Jan 8, 2021 at 16:49
• @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.