Re: best braking technique as one approaches red light

I'm traveling at a speed V when I see a red light
ahead at distance d with probability to change
to green as function of time p(t), which for example
could be gaussian.
I would like to find the velocity profile v(t)that
maximizes my velocity (averaged over all scenarios
knowing p(t)) as I pass the light, with the
conditionthat I have to stop if I hit the light when
its still red.
It doesnt seem to fit the form of a standard
functional cauchy-riemann type problem. Has anyone

I meant here Euler-Lagrange

insight how to solve it?

If you absolutely have to stop on red, then the
probability function
p(t) is irrelevant. You always should go at the
maximum speed subject
to the requirement that you must be able to brake to
a full stop at
the light if it's still red. - quasi

No, the probability is relevant. If for instance
the probability of being green is:
p(t)=0 for t<5 and
p(t)=1 for t=>5
then I should go the speed d/5.
The 5 in d/5 comes from the probability function.

Maybe I'm missing something, but it seems to me
that you can't possibly do better than to go at the
maximum possible speed for which it is still possible >> to stop. And there's no reason to do worse.
What makes the probability function irrelevant is
the fact that you
are not allowed to fail to stop if it's still red.

Ok, I just noticed -- p(t) can be zero on an
interval.
Perhaps that does make p(t) relevant.
I have a few questions (although I won't be back
until later):
p(t) represents the probability that the light will
go green within t
seconds, right?
Does the function p itself change based on how long
the light has been
seen as red?

p(t) can be taken as gaussian for instance.
It does indeed change with time t.

Lets call big P(t) the probability that light is green at time t, this being integral of prob density little p(t)dt that light changes during interval [t,t+dt]
Look at my example again if you do not get why p(t) is relevant. Do you agree it is relevant there, or no?

V is the maximum speed, right? But what about the
maximum acceleration
and deceleration values? Don't we need to be given
that info?

Take either a maximum accel a or infinite a, whichever makes problem easier.

Do these lights have an amber stage?

Haven't done the sums but it could be that you would
slow as you approach
the lights and then speed up knowing that it they go
amber you will still
get across before they go red. (Assuming that you
know how long they stay
amber.)

Lets say its US system where red goes to green with no amber. I agree the best approach will be some amount of slowdown to increase the probability that the light is green when you hit it. But not too much slowdown since we want to maximize speed on hitting green light. It is a 'good' problem since these two factors both play in opposite directions. If you hit the light when its red you must go to 0 speed, lets say with max acceleration a, or if it makes the problem easier with infinite acceleration.

If I had a velocity curve that gives me 20km/hr on hitting
the light, and this velocity curve gives a 10% chance of hitting red, then I get the average of 18km/hr as result.

And lets say that P(t) is probability that light is green at time t, this being integral of prob density p(t)dt that light changes during interval [t,t+dt]
.

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