I have measurement data which looks like the sum of an exponential and a constant function. It is enough to see it as a continuous function, say $f(x)$.
I am looking for the $a$, $b$, $c$ values for which
$$\int\limits_0^1\big(ae^{bx}+c-f(x)\big)^2dx$$
minimal.
I think the problem is well-known, although typically such functions are approximated with polynoms.
What I did:
First I've tried to find the local minimums, so I created an equation system using
$$\frac{d}{d\{a,b,c\}}\int\limits_0^1\big(ae^{bx}+c-f(x)\big)^2dx=0$$
The result was a practically not calculable equation system, after filling some sheet of paper with formulas. Using the Lambert W wouldn't be a problem for me, but the situation was too complex to apply even that.
However, such a well-known and probably useful problem probably has already some results.
Actually, even a local minimum would be already useful, I don't suspect multiple local minima in the $(a,b,c) \in \mathbb{R}^3$ space.
Or, maybe an entirely different direction should be used?