I have been reading about optimal control and in the various classic examples they usually come with the built-in cost function. But I have a question under what criteria was built that cost function ?, or what is the methodology that follow to find this functional ?.
So, if I have the following model of car-following with delay
$\ddot{x}_n(t+T)=\lambda[\dot{x}_{n-1}(t)-\dot{x}_n(t)], n=1,2,3,....,k,\quad T>0\text{ fixed}.$
I was working with the equations of the movement of a vehicle considering friction, where I got the next functional,
$J(\alpha(t))=-\int_{t_0}^{t_f}\frac{\mu}{m}\alpha(t)dt$,
where
$\mu:$ Coefficient of friction
$m:$ Mass of the vehicle
$\alpha(t):$ Velocity (the control function)
In general, Under what criteria can you build the cost functional that minimizes travel time?
How can I associate the characteristics of the road (narrow way, wide way, presence of curves, etc.) for the construction of cost functional.
Merci beaucoup!