Your microphone is off.

Okay.

Sorry for that.

So hi everyone and welcome to the seminar. So today we have a professor of Francisca Rossi from the University of Padova in Italy.

And today we're speaking about the control and regularity of non-local transport equations.

Please, Professor Rossi. Thanks a lot. Thanks, Marius. Thanks, Erika, for the very kind invitation. It's a strange way of presenting, but it's very, very interesting.

And thank you. It's sad I cannot be in Germany. That would be much better. Anyway, my talk, as you see, deals with the control and regularity for non-local transport equation.

For this reason, I will start by saying why we care about non-local equation, what they are, and why we use them to model a multi-agent system.

I will then focus on two articles that come from collaborations with a collaborator in France, Michel Duplem or Gamoran for the first part, and Benoit Bonnet, my former PhD student, for the second part.

The first will be about controllability of equation, linear equation, by the way, not non-local, and the last part about the regularity of optimal controls in a local transport equation.

So let's start by saying something about multi-agent system. When you think about multi-agent system, we think about a large number of agents that interact.

The first one in history was the study of road traffic. We have a lot of work about pedestrian and the non-local equation are very useful to describe the pedestrian.

There are applications in robotics, in particular in network control, so a large number of small robots that are able to interact and perform complex tasks, even though each robot is quite simple one.

We have studies in biology, for example, in these very beautiful flocks of birds. By the way, this is the season, at least here in Italy, we see a lot of flocks here around in these days.

And finally, I will not describe my researches in more biologically oriented problems, but we can even think about human or animal tissues as crowds of cells that interact by sending signals, by sending proteins, by the way.

So if we think about what we want to do when we model multi-agent system, we want to describe the evolution of a large number of agents that arises due to the interaction between them.

For example, between a small number of neighbor agents that interact and give rise to global behavior.

And an interesting aspect that is very useful, and this is the reason why we use the local transport equation, is that they can unify some models that plays in which there are agents with different importance.

For example, in the main field games, you have the word major players, or even in crowds, you have this structure of leaders and followers.

If you want to describe how an agent behaves inside the crowd, the very basic idea is to say, I look around myself and try to choose the best trajectory, for example, to get out of a room.

And the mathematical language that is most suitable for this goal is to use a non-local velocities. What does it mean? The main idea is to say, OK, I have an area around myself that is well described by a convolution, and I decide my velocity as a convolution between a given kernel and the crowd around myself that is described by mu.

The main idea is that I look around the crowd and choose the best option. OK, this idea can be written as follows.

As a vector field V that depends on mu, so depends on the whole crowd, not only on the value in my point, but around, and I choose the value of this vector field in my point.

This means that the operator is an operator that takes the whole crowd mu and gives you a vector field.

OK, as I said in our work in the main field games, for example, this process of choosing the direction comes from a minimization problem.

OK, this is somehow hidden in what I'm showing here. We will say just that at each configuration of the crowd, the mu at each point takes, we have a velocity and that this velocity is very regular somehow.

So we do not consider the minimization of the cost at the agent level. We will see that we discuss control and optimal control at the global level. OK, so as I said already, we have leaders and followers in crowds often.

So this is good to use to describe a crowd as a measure, for example, a measure that contains a continuous part. Let's say most of agents are considered like a fluid. OK, it's a very important discrete agent that can be described, for example, by Dirac Delta.

And so the best way to do this is to work in measures. And we have a lot of models in the mathematical community, for example, by people like the GOM, Moritz and Ambroggio, Piccoli, Tosin, and so on.

And here you can see an example of a simulation based on a multi-scale model in which you have a group of people here.

You have a leader that is at the top. The leader moves to the right and then you have something important happening here is that you have a queue that arises in a very natural way.

So one to one interaction between people gives you a queue that arises as a global shape.

And the interesting fact here is that if you think about non-local equation, you might always wonder if it is good to deal with a discrete or continuous part. OK, why? I mean, in some sense, one might say, OK, a crowd is never continuous. OK, this is an approximation.

OK, and in reality, the key point is that here we use the fact that you can pass very easily from discrete to continuous description of the crowd by using the so-called mean field approach.

So let's think that we have a very large number of agents, X1 to Xn, that are here. They interact one by one by a function. Here you see I'm in a discrete setting because I'm writing an ODE.

And let's say that since they are all identical, I can just write the measure associated to the solution to the ODE. So what happens to the sum of all the agents that interact?

And the question that we say is that, OK, let's start from an ODE in which all the agents are identical.

And let's try to see if mu, muon satisfies a PDE. OK, this is a measure. So this is a PDE in the space of measure. So in the weak sense.

OK, and it is very well known. Sorry, I don't know if you can see the last part of the screen. This is a shame. OK, OK, this is good.

OK, something that is very classical is that if f is sufficiently regular, for example, Lipschitz, then it's easy to see that the right choice of the vector field is just f convolution with mu.

And so in this case, you can have a complete connection between the discrete and the continuous part.

You just decide if the measure is discrete, you have ODE. If the measure is continuous, you have a classical PDE, non-local.

If you have a measure that mixes both, then you have no problem. This is the key idea behind most of the reasoning I will show you today.

And OK, so let's go. OK, the question that one has when writes a new PDE somehow is to see if there are there is any problem about

existence, uniqueness and continuous dependence. OK, I will be very short on this because this is not something that will be central in the presentation.

Let's say that everything works well. OK, everything works well in the following sense. If we consider the Cauchy problem, starting from a measure mu 0 and moving

with a transport equation that is non-local, everything has existence and uniqueness.

You have existence and uniqueness of the solution in the following case. For each V mu needs to be a Lipschitz vector field, let's say with a bounded Lipschitz constant,

a fixed Lipschitz constant, you need to be bounded in an infinity. You can even be sublinear. There's no problem.

OK, but the key point is the following. You need to be Lipschitz as a function of measure with respect to the Vass-Szstein distance.