I am trying to solve an inverse problem based on variational principle.

I will first present a forward problem that is already solved, and then present the inverse problem that I am trying currently to solve.

The forward formulation of the problem is the following:

Given a known function $f(x)$, an unknown optimal deterministic trajectory $z_m(t)$ is defined by a variational principle $$ \delta \int\limits_a^b L(\dot{z},z) dt = 0 $$ with $z_m(a) = x_0$ and $z_m(b) = x_f$.
So $z_m$ is the minimiser of the Lagrangian $L$ that involves the function $f(x)$, and we can formulate Euler-Lagrange equations $$\DeclareMathOperator{\D}{\operatorname{d{\!}}} \frac{\D}{\D t} \frac{\partial L(\dot{z},z)}{\partial \dot{z}} = \frac{\partial L(\dot{z},z)}{\partial z} $$ to identify $z_m(t)$ for $t \in [a,b]$.

The inverse problem I am trying to solve:

If the function $f(x)$ is unknown but optimal deterministic trajectory $z_m(t)$ that is the minimiser of the Lagrangian $L$, is known, I want to find a way to identify $f(x)$ in a functional form.

The Lagrangian (whose $z_m$ is a minimiser) is of the form $$ L(\dot{z},z) = \frac{1}{\sigma^2} \big( f(z) - \dot{z} \big)^2 -\frac{\partial f(z)}{\partial z}, $$ so it involves both the function $f$ and it's 1st derivative ($\sigma$ here is some constant).

From the Euler-Lagrange equations I can obtain an equation that involves $f$ and its derivatives evaluated at the known optimal trajectory $z_m(t)$.

So in essence I have an equation of the form $$ \ddot{z}_m(t) = \frac{\sigma^2}{2} \frac{\partial^2 f(z_m(t))}{\partial z^2} + \frac{\partial f(z_m(t))}{\partial z} f(z_m(t)). $$ Since I know $z_m$ in a functional form, I could evaluate $z_m$ and its derivatives within the interval. However, since the equation that I obtain for $f(x)$ contains both $f$ and its derivatives I do not know how to proceed, or whether this problem is solvable at all.

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    $\begingroup$ Probably not needed but wanted to give the general setting. Thank you for the feedback! $\endgroup$ Aug 10, 2022 at 14:10
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    $\begingroup$ Maybe googling on the viscous / viscid Burgers-Hopf equation might give a lead on a method of solution. See, e.g., "Burgers equation" by Mikel Landajuela bcamath.org/projects/NUMERIWAVES/… $\endgroup$ Aug 10, 2022 at 17:50


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