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.