I recently encountered the following optimal control problem.
The purpose of the system is to find the parameter $x$ at which the maximum or minimum of the function $f$ a is reached. $x$ is unknown to us in advance.
We have gradient type PDE:
$\frac{dx}{dt}=\frac{df}{dx}+u$
where $f=\frac{1}{(x-x_*)^2+1}$, and $x_*$ - constant, at which the maximum function is reached.
Problem: Set the cost function $J$ so that the transient process from $x(0)$ to $x_*$ in the system is exponential, i.e.:
$J = f(x,x^{'},x^{''},u) =?$
I am new to optimal control of partial differential equations. I can't seem to figure out how to find an approach to this problem, so any help would be appreciated. How to set the cost function $J$? How to generate an input control signal $u$? Do we have an analytical solution? I thank all the helpers.
