So, in the last lecture, we investigated this innocent-looking model or system,
which was a ball bouncing on a surface. And we saw something rather strange
there, which was that, that the, the ball ended up bouncing an infinite number of
times in finite time. And this is part of another potential
complication that comes from hyberdizing your model, namely, that you have these
kinds of infinitely many swicthes. And this is known as the Zeno Phenomenon.
And in today's lecture, we're going to dig a little deeper into the, the Zeno
Phenomenon and see what we can do about it and if you can understand it.
But fundamentally, what I would like to point out is that Zeno is bad, because if
you're actually running something that's asked to do an infinite amount of things
in finite time, it crashes. If you're running this on the computer,
the simulations crash. another thing is that we know that there
is something inaccurate or wrong with our model because the ball, if I drop a ball,
it doesn't bounce an infinite number of times,
it bounces 17 times and then it stops bouncing.
So, there's something wrong with our model.
That's another warning flag. And the third warning flag is that we
don't actually know what the system does beyond the, the Zeno point, meaning the
time up to which we have an infinite number of switches.
So, since we can't really define what the system is doing beyond that point, things
like asymptotic stability is meaningless because time is not allowed to really
progress off to infinity. So, first of all, why is it called the
Zeno phenomenon? Well, there was a Greek philosopher, Zeno, Zeno of Elea who spent
a lot of time thinking about movement and the dynamic world and basically his point
was that our perception of the world is wrong because clearly there are all these
problems out there. For instance, here's one of his famous paradoxes.
We have a hare racing a tortoise. And the tortoise is a little slower so the
tortoise gets a head start. In fact, the tortoise starts there and
then, the race is on. And at some point, the hare reaches the
point were the tortoise started from but at that point, right, the tortoise has
moved, not much but it has moved a little bit.
This is how far the tortoise has moved. Okay. The race goes on.
And at some point, the hare catches up to where the tortoise was last time but now,
the tortoise has moved a little bit more, not much, and then this repeats.
In fact, here is the, the paradox. The paradox is that the hare never catches up
with the tortoise because every time it reaches the step that the tortoise was
last time, the tortoise would have moved a tiny bit.
Now mathematically, this is nothing.
We know now about convergent series. We know that even though there are
infinitely many of these small intervals the sum of them will converge and there
is indeed a point where the hare will catch the tortoise.
but the problem for us is that if I model this as a hybrid system, I have, again,
infinitely many switches in finite time. So, this is why this kind of infinite
amount of switches is called the Zeno Phenomenon because it can be traced back
to Zeno's many paradoxes about motion. Now, let's look at another example, one
that's not a hare and a tortoise but one that's rather innocent-looking.
