Optimising closest approach with third order motion

[ucclaw]

I have recently stumbled into this problem trying to visualise a certain economic model and I'm finding the solution is just beyond my reach. As far as I can tell there is a simplification of the problem which is easier and would still be good to have answered.

There are two moving point particles A and B on a Cartesian plane. Particle A is trying to reach B as quickly as possible, it can do so by applying acceleration in any direction. Its acceleration and speed have constant upper limits. Particle A knows the position and velocity of itself and of B, and can continuously[1] adjust its acceleration (which doesn't have to be continuous). What function of the particles' position and velocity should A use to reach B in as little time as possible?

In the simplest version B is moving with a constant velocity, which I think will produce a better result for the complete version too, in which B can have acceleration. For many nodes targeting each other my current approach of "accelerate as fast as possible to B's current position" quickly resembles chaos for a few particles targeting each other in a chain. I know why that's a poor approach, I just don't know how to make a better one.

If I haven't been clear enough, I'd be happy to provide a visualisation of the problem and my not-working solution.

EDIT: Pretty please work this all the way through. I'm here because I've shown a lot of people this problem and the pattern has been for them solve it for a single dimension then tell me that it should be easy to just do it for both dimensions. It really isn't, there's almost certainly a derivation and optimisation equation in there somewhere since there are three unknowns: acceleration on each axis and time but only two equations to solve them with (position of both particles with respect to time). Another way of looking at it is that the dimensions can't be considered separately assuming they can move with maximum acceleration, since the magnitude of the acceleration is limited there is a trade-off there. As is probably clear from my fumbling explanations, I'm not math savvy enough to translate this idea into number and figures.

I'd be extremely grateful if anyone can help with this, I've spent countless hours now trying to solve it and I think I'm just not experienced enough.

[1] Not actually continuous, since it runs on a computer in discrete (but tiny) steps.

 

[Stephen Tashi]

To look up the answer to your problem, my guess is that the mathematical search phrase "calculus of variations" should be used. For example, if you search on:
calculus of variations, intercept course
you find literature on how to guide missiles to hit targets. That isn't quite what you want since missiles have inertial, which constrains them.

You can also try searching on "pursuit problem", which gives hits for the special case that you asked about: http://mathworld.wolfram.com/ApolloniusPursuitProblem.html

If you need more help, let me know and I'll actually think about what you asked!