Okay, so today I will start discussing an interesting problem which is the localized

damping.

So why I'm doing so?

Well, yesterday we were considering the wave equation y t t minus laplacian y plus y t

equals zero that we have written in this manner, right?

So that we can understand that the feedback is being applied everywhere.

This is definitely a limitation in the context of control problems, right?

Because this means that if the membrane or the acoustic waves propagating are occupying

this domain, this will mean that the control is acting everywhere inside.

So this will mean that the whole domain, right?

The whole domain will be full of actuators, right?

We need indeed, you know, an actuator at every point x being able to apply a force which

is precisely the velocity y t at that point x.

So this will mean actuation everywhere, everywhere in the domain, in the domain omega, meaning

that at each x we have an actuator able to realize, you know, the force F x t which is

equal to minus y t x t.

Now, if you talk to, you know, practitioners of this kind of damping mechanisms, you will,

you know, or sir, they will tell you, well, listen, this is not very realistic.

You never have, right?

You never have actuation everywhere in the domain, right?

So there are many, many applications of these kinds of problems.

There is one that we often experience and we don't even realize it.

Sometimes they are not very visible, right?

So we can see that on highways, right?

So on highways, for instance, we observe that especially when new highways are built, right,

oftentimes, you know, on territories that are occupied by, you know, surrounded by

residences, we often observe that there are barriers, acoustic barriers that are built, right?

So these acoustic barriers can be sometimes solid walls and some other times they are not even solid.

They are just simply some piecewise constant material columns that are, you know, built so that the noise

generated, right? So the noise generated by the cars when traveling, when getting to the border of the highway

is reflected inside, right? So that's the idea.

So this is a typical example where, you know, the damping, right?

The damping is localized on the boundary of the domain, right?

So this is our first observation.

You see, in this case, someone could say, well, what is the relation of the wave equation with this problem of noise

propagation, noise generated by traffic on highways?

Well, of course, the wave equation is the model also for acoustic waves, right?

So the pressure field, the pressure potential of acoustic waves is modeled, at least in the linearized

regime, by the scalar wave equation.

And the problem of noise reduction, as we have mentioned in the early lectures of this series, is one of the main

motivations, main applications of control theory, independent of what noise means, right?

So noise can be acoustic noise, but it could be also the noise on a blurry image.

OK, so then you see that in this particular example, that it is natural to think of damped waves,

damping noise, absorbing noise.

But when you do so, the damping term, the dissipative term, is only localized on the boundary, is not put everywhere inside.

So here, actually, we need, right, this space completely free so that cars can travel.

We cannot claim that we are, say, embedding some kind of damping mechanism within this medium along the highway.

The highway needs to be free so that cars can travel with maximum visibility and without any obstacles, so to make traffic safe.

OK, so the damping is on the boundary.

This means that, you know, in particular, it will be very natural to consider situations where, you know,