Optimal control of the gradient type PDE

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.

$$\frac{dx}{dt}=\frac{df}{dx}+u$$

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.

• 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? Commented Jan 15, 2021 at 14:41
• 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 $a>0$? Commented Jan 16, 2021 at 21:55
• @fibonatic please, see my edit and help me
– ayr
Commented Jan 17, 2021 at 5:42
• But with exponential transient you do mean $x(t)=e^{-a\,t}(x(0)-x_*)+x_*$? Commented Jan 17, 2021 at 16:51
• @fibonatic Yes!
– ayr
Commented Jan 17, 2021 at 16:52