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I'm interested to learn about how discrete/'event' based elements are incorporated into optimisation problems. Hopefully this is an interesting problem in its own regard, it's inspired by a daily occurrence in my life.

The problem stated in english is:

What is the optimal strategy to unlock my car with its remote, soonest, as I walk towards it?

The function that determines when the remote will be able to unlock the car is dependent on the battery voltage $V$ and the distance I am from the car $d$:

$$f(V,d) = \alpha \frac{V}{d^2}$$

  • If this value goes over some threshold $\tau$, the car will unlock.
  • $\alpha$ accounts for some fun (but unknown) stuff like the permeability of air and so on (let it be 1 if you like).
  • I am walking towards the car at a constant velocity $u$, from an initial distance $D$.
  • The car is not unlockable at $D$.
  • The initial voltage $V_0$ is positive but unknown.
  • After each attempt to unlock the car the remote the voltage $V$ is reduced by a multiplicative factor $\{\beta\in \mathbb{R}\mid 0 < \beta < 1\}$:

$$ V_{n+1} = \beta V_{n}$$

I think that's about it in terms of required information...

(please comment if not, thanks)

Can an optimal schedule (strategy over time) of unlocking attempts be defined, given no initial values?

In order to avoid making the question too broad, the following aren't specific questions, but 'discussion points' :)

  • Obviously if we're given enough values we can figure out a $d$ and use one optimal attempt $(V = V_0)$, but it's more interesting having to account for the Voltage loss in an unknown setting (right?).
  • In general, how are non-linear thresholds like $\tau$ and the event driven schedule of attempts dealt with in problems like this?
  • Is there a difference if the problem states that the complete schedule must be derived prior to the 'walk'? ...That is, compared to adjusting the schedule dependent on new information such as 'the last attempt didn't work'?
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