Now we should be able to share screen.

You should see my presentation now.

Yeah, now it's there.

Okay, let me.

Now it is in full screen, no?

Yeah, it's in full screen.

Okay, perfect.

Well, okay, first of all, thanks to the organizers of these online sessions.

Thank you also for all the work of organizing all these nice seminars in this period of

lockdown.

And okay, today I'm going to present a recent work that we very recently submitted some

weeks ago in collaboration with Anna Navarro, who was a postdoc in our group in Bilbao.

Now she's in the University of Valencia.

And it was also well, the title of the paper in the work is based on is the same one of

the talk.

You can find it on archive if you're interested in more details.

And well, the topic is essentially the application of stochastic optimization methods to some

simultaneous control problem for parameter dependent systems.

Okay, so essentially in this talk, there will be three concepts, which are the one of parameter

dependent models, the one of simultaneous controllability and the one of stochastic

optimization.

And so I will just briefly introduce the three of them in the next slides and then we will

go a little more in details.

Well, okay, parameter dependent models actually, there is nothing mysterious in them.

They are models that can be in a general form given by these kinds of equations.

In their A and B can be linear, can be either matrices or operator linear spaces.

So we can be talking about all these or these, in the talk we focused on the ODE setting.

And they are depending on some parameters that we indicate with new, which essentially reflects

some changes due for instance, for instance, to external factors.

If you consider this model can be for instance, some model in climatology.

The behavior of the solution depends also on some external factors, which can be the

pressure, the humidity.

Well, some example, I have here a couple of examples.

One just in the linear setting is the one that we will use after then also for the simulations

is this simple four dimensional ODE, it's a linearized version of the Cartinverted Pendulum

System.

And here the parameter mu will be the mass of the pendulum.

You see the purple dot here in the figure.

Or if you want to go in the PDE setting, where there is for instance, this very famous model

of thermo-elasticity, where, which is essentially is a model well known in the material sciences

and it is well known that its behavior depends on a couple of parameters that are actually

famous they have a name, they call them coefficients, which essentially describe the properties

of the material that is the model by the simulations.

And we talked about controllability for parameter dependent systems that are essentially two

well defined notion of controllability.

One is the one of average controllability that also, in fact, John has already mentioned

in his talk before, but we are not going to talk about it.

Essentially the idea is that you want to control the average of the states with respect to

the parameters.

But what we are interested in in this work today is the concept of simultaneous controllability,