Welcome everybody.

And I would like to describe to you some mathematical modeling and some analysis of these models

related to multi-agent systems.

And in particular, we will try to explore a little bit a couple of concepts, the concepts

of sparse controllability multi-agent systems.

And for that, what I mean is that if you have multiple agents that are interacting, the

problem that we are going to approach is if I want to induce certain configuration of

the entire group, and I probably don't need to interact with everybody in the group, but

I need only to influence a few key players in the group, and still I'm able to kind of

steer the dynamics.

That's why I call sparse the control.

So it's a control that is zero for most of the agents and is acting on very few of them.

This is the first concept.

Second concept is the so-called mean-field optimal control.

What does it mean?

It means that when the number of agents is so large, we might prefer to analyze the control

of the probability distribution of the agents rather than looking at what happens at the

particle level.

So that's another concept that I'm going to discuss.

If I have time, and this is possibly not the case, but if I have time, I will talk about

the learning of the system.

So let's assume that we have a multi-agent system that we can observe, so we can observe

the dynamics of it.

We would like to learn the interaction rules within this system, so the kind of rules that

are governing the dynamics.

That's something that I will talk about if I have a little bit of time.

So which kind of phenomena do we want to model?

So here I give you some examples.

So we do have the behavior of multiple agents.

For instance, birds, people, traffic, full of fish.

So these are, if you want, examples of social dynamics.

And one way of approaching the modeling of these kind of behaviors, in all the examples

that we saw, they were examples of motion, right?

It was a motion of birds, motion of people, motion of traffic, motion of fish.

And in all these cases, somehow, one could try to borrow a leaf from physics and try

to describe the system in terms of a Newtonian type of description, right?

So that's the kind of idea that we might consider here.

So of course, this is not the only kind of modeling one can use.

Yesterday, in the seminar of applied analysis, I presented sort of evolutions driven by games.

And so certainly, this is not necessarily the right or the best or the unique way of

trying to model social dynamics.

But we can do it, and we can embed into the dynamics certain kind of social forces that

are inducing a change in the motion, for instance, of certain agents.

Now I describe the action of a force, in particular the force K, not only particle-wise, but in

particular group-wise.

And so as a sort of binary interaction rule, and we can describe this in terms of a convolution

of force in kernel with respect to the empirical probability measure that represent the agents.

In particular, we can represent the agents as deltas pointed in certain positions with

certain velocities.

So that's the kind of model that we might consider, because we would like to model the