There are many practical applications in which there is not just one actuator, but two, right?

So in this particular case, for instance, I am just to give you an example, I consider

the situation where the dimension of the system is capital N. I'm sorry, I changed

notation a little bit from the previous slides where it was small n, but this is a finite

dimensional system. Of course, everything I'm talking here, you can do it also in the

infinite dimensional case, right? But then it's more technical in particular because

you have to discuss the different kinds of semi groups you are, when it's generated,

And then I am considering the situation where you have two actuators, right? So b1 and b2,

so bj will be vectors in Rn, right? So these are like two compressors, two engines, two

actuators that you can manipulate in order to guide the whole system. As I said yesterday,

one of the analogies in the context of machine learning is the federated learning, right?

Where two different groups of people are cooperating in order to generate, for instance,

a neural network that is able to interpolate and generalize well data on a given topic

on which each of them have partial information, right? So you, for instance, try to discuss

behavior of people. One person has data according to gender. Another group has data

according to ages, right? And then both of them are trying to build a neural network that is trying

to anticipate the behavior of people, both according to gender and age, right?

So here is a little bit the same. You could say, well, so here now I have two controls, right? So

I have two controllers, two actuators, b1 and b2. Each of them is acting differently on the system.

It could be like, you know, like two laser beams acting in different directions, and you can just

regulate the intensity of each of them. So the controls are u1 and u2, right? Both depend on time

and both are scalar. So when you look to the problem that way, you say, well, there is nothing

new on it, right? Because I could simply say that these, all this, right, I could write all this

as being, right, as being some b and then some vector u where the matrix b now is n times two,

right? And it's a matrix with two column vectors b1 and b2. And then the control u tilde will be a

vector with the two scalar controls, u1t and u2t, right? So then you say there is nothing new here.

You could simply, you know, apply Kalman rank condition and determine whether a, b fulfill Kalman,

and that will be the condition for controllability. Yeah, that is true. But in many practical

applications, right, you can be in such a situation in which you don't want to use both

controls simultaneously, right? So you like to use first one and then the other. So for instance,

this is what typically on hybrid cards, hybrid cards you expect to do, you use either gasoline

or you use electricity, right? Okay, you can, of course, use a combination of both, but let us

consider the, the extrema case in which you either use one or the other, right? So this is like

when you are playing ping pong or tennis, so one kicks and then the other one kicks, but not both

of them are kicking simultaneously, right? This is a typical application. So the question is then,

right? The question is, the question is, even if a, b in such a context is controllable,

can we control the system

system? So that

u1t times u2t is equal to zero, right? So what does it mean? The fact that u1t times u2t is equal

to zero. It means that either u1 or u2 or both are equal to zero. So it means that you never,

in any time instance, you are not using the two controls simultaneously, right? You are either

using one or the other or simply doing nothing in case that, you know, is there is no need of action.

Okay, so this is a switching control system because I am switching from u1 to u2. And as I said,

in the context of reinforcement learning, when, you know, two different groups are

cooperating in order to build, you know, a neural network that is interpolating and generalizing

data properly, right? You want this to happen also in that alternating manner, in that switching

manner, in particular, because they are not allowed to hybridize to mix data, right? So each control

has to act on the system with information the control can perceive, right? Okay.

Good. So in practice, switching systems appear in many different applications. So for instance,

in many applications, you know, like medical, when a human takes a given drug, right?