# I'm walking towards my car - when should I try the remote, in an optimal sense?

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...

• 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?