Optimal control model of an articulated part that starts from station A and arrives at station B.

Question

I have the following optimal control problem:

Formulate a model of optimal control of a joint that starts from station A and arrives at station B and you want to know how far you should start to decelerate and you also want to know the optimal time.

1. I have an articulated bus that moves in an exclusive way and part of stop A with initial velocity equal to zero and arrives at stop B with final velocity equal to zero and the speed limit on the road is $40 km / h$.
2. I am considering that the traffic lights that are in the exclusive way are in green light every time the articulated bus is near.
3. The driver of the articulated bus is interested in knowing at what distance he should stop accelerating the articulated to reach stop B with zero speed. With this we are indicating that we must control is the acceleration due to the accelerator, that is, the acceleration will be my control variable.. From here I got the following question: How to find an equation of the speed that is a function of the position ?, that is, $v = v (x)$ and also how this variable could add variables such as speed in curves, if the road has slopes, among others.
4. The maximum acceleration I think will be limited by the capacity of the engine and the maximum deceleration will be limited by the capacity of the braking system.
5. Finally, How can I formulate a cost functional that minimizes travel time?

Answer 1

The basic equations for constant acceleration with $s$ being position, $v$ being velocity, and $a$ being acceleration are $$v=v_0+at\\s=s_0+v_0t+\frac 12at^2$$ with the zero subscripts representing starting values. If you have a limiting velocity $v_{lim}$and fixed acceleration, you reach that velocity after $\frac {v_{lim}}a$ at distance $\frac 12at^2$. That is the distance from the end you should start to decelerate at.