Functional inverse problem based on a variational principle

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.

• Probably not needed but wanted to give the general setting. Thank you for the feedback! Aug 10, 2022 at 14:10
• 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/… Aug 10, 2022 at 17:50