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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.

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    $\begingroup$ This doesn't look like a PDE since $df/dx=-2x/(x^2+1)^2$. And I also don't see how optimal control and objective function come into this, or can $f(x)$ be chosen to be any function? $\endgroup$
    – fibonatic
    Jan 15 at 14:41
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    $\begingroup$ could you also clarify with what you mean by choosing the objective function? Do you mean choosing $J(u(x))$ such that its optimal solution yields $x(t)=e^{-a\,t} x(0)$ with $a>0$? $\endgroup$
    – fibonatic
    Jan 16 at 21:55
  • $\begingroup$ @fibonatic please, see my edit and help me $\endgroup$
    – dtn
    Jan 17 at 5:42
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    $\begingroup$ But with exponential transient you do mean $x(t)=e^{-a\,t}(x(0)-x_*)+x_*$? $\endgroup$
    – fibonatic
    Jan 17 at 16:51
  • $\begingroup$ @fibonatic Yes! $\endgroup$
    – dtn
    Jan 17 at 16:52

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