[kubark42]

Originally Posted by tasdrouille

Building on Robert's idea, of letting it start backwards with the clutch disengaged. You now have two scenarios. 1- The car reachs a speed at the bottom of the wave at which the engine would rev past its efficiency island. 2- The car never makes it to the most efficient speed just rocking back and forth from it's innitial potential energy.

Scenario 1: You let it rock backwards up the left hand side of the wave. From there going down it will cross the most efficient rpm somewhere before the bottom of the wave. It will also cross it back while slowing down somewhere on its way back up the wave to the car's original starting point. Assuming your new "starting position" is now on the left hand side of the wave pointing down, you then calculate and subsequently execute two pulses, both as close to the efficiency island load and rpm as they can be, one on the way down and one on the way up, that will give you the energy needed to get over the first wave.

Scenario 2: You let it rock backwards so your new starting position is on the left hand side of the wave pointing down and you do one single pulse starting at the very bottom of the wave, where you're speed will be the greatest.

Edit: Thinking about it, for scenario 1 the car should be rocking back and forth in the starting wave while applying infinitesimal bursts each time the sweet spot speed is crossed when the car is going forward, the amplitude of the movement increasing all the way untill the car can go over the first wave.

That is indeed correct. I think an easier way to look at it is like this:

You need a certain amount of additional energy to climb over the first hill. The maximum efficiency will come by buying that energy the cheapest possible.

Since the car can rock back and forth for free, you have only to let it oscillate back and forth, applying thrust only at the moment when the energy is cheapest to buy, that is to say at the highest efficiency.

So to be mathematically precise, you have the following cases:

1) While coasting up and down, you never reach the maximum efficiency speed. At this moment, you apply an infinitely short blip of throttle in the very bottom, buying your energy at the maximum efficiency point, even if it is not the absolute maximum efficiency.

2) While coasting up and down, you reach the maximum efficiency speed. You can do this once (in the critical case), twice, or perhaps an infinite number of times.

2a) Once: The maximum efficiency speed arrives at the very bottom of the curve. This is an arbitrary distinction from the multiple peak crossings, so there's nothing interesting here.

2b) Multiple times: depending on how strong your torque input is versus the slope, you could actually add energy sufficiently quickly that the car accelerates. Therefore you get to repeat this strategy as many times as possible during the cycle.

In any case, the strategy is always to add energy at the very moment when it costs you the least. And an earlier comment was right, once you have just enough energy to crest the first hill, the other hills require no more energy, but you WILL grow tired of this after a few eons and decide that maybe a tiny bit of wasted energy might be nice in order to finish the trip before the universe dies a heat death.

I have long been suspecting the definitive answer is lying in an unpractical particularity of the excercise, mostly irrelevant to the real world.

That's not correct. It is, perhaps, not immediately relevant to driving. However, it is most certainly not irrelevant to the real world.

One of the challenges in the world of controls and mathematics is to find analogs between systems. For instance, you have developed a new controller and would to apply it to nuclear reactors. Obviously, no one is going to let you do it without years of experience with the controller. So how do you do it? You look for systems with similar dynamics. For instance, the ball and plate is analogous to certain nuclear controls problems, so by demonstrating stability in a fun, harmless experiment, we can at the same time make the case for a far more serious application.

So, to come back to the point, what is interesting is that there is a point at which the system "tips", at which it is no longer interesting to go backwards an infinite number of times before we finally arrive at our forward target.

So while this might not work in a real-world car, something similar might work on metal rails (with perhaps one or two rocking motions before the final "push"), certainly works for spacecraft (that's where the "intersteller highway" comes from), and probably has some interesting applications in chemical systems.

Anyway, all that over and done with, I thought it was an interesting problem when I encountered it, and it highlights the problems with correctly posing problems in optimal control.