And also for organize these mini workshop.

I also I would like to thanks and because this is my, almost my last week here in the university I come back to chill the next Wednesday.

I would like to thank Enrique for inviting me to come here for two months and work with Enrique and all the collaboration that we do,

for the discussion in maths, politics also and football, for the couple of glasses of wine that we drink in this state.

So thank you Enrique for everything. It is always a pleasure to come here and visit and work with you. Muchas gracias Enrique.

And also I would like to thank all the members of the data science department here in the University for your kind hospitality during my stay here.

So the talk. So my talk is the Controllability for the viscous and brain hardened Enrique equation.

This, this work is part of the thesis of Pablo Marquez from the same University, University of Santiago, Chile, to obtain the degree of mathematical engineering.

Pablo defended his thesis in July of 2021. So almost all the talk that I give you, that I will give you, is part of the thesis of Pablo.

And also my colleague Galina Garcia from the University of Santiago was the co advisor of the thesis of Pablo.

So the talk, I divide the talk in four parts. First, a little motivation to study this model and one important property that we see in a few minutes.

Then the wall position of the equation and some properties, some spectral properties for the operator that generated the system.

Then the main result of the thesis of Pablo, the controllability result, we study the null, exact and approximate controllability for this equation.

And finally, a work in progress that is done with Galina Garcia.

That is the numerics for the controllability result that we do. And one open problem that maybe is interesting for one of you.

So the motivation. In the 40s and 50s, the interest in studying the propagation of pressure waves of a small amplitude in bubble liquids appeared.

Why? This is the question.

The reason was to determine whether it was possible to take advantage of these acoustical properties to control the sound produced by propellers.

Of a ball of surface chip and submerged chip.

So what is the meaning of that?

So it is hoped that the submarine were not detected by the sonar of enemy chips.

So this is the reason that the mathematician and physics study this propagation of pressure of a wave of small amplitude.

So it is for our point of view.

So we have a lot of literature about the system. I present one of them.

The one dimensional flow of liquids containing a small gas bubbles.

This is a very nice paper, very well written. So if you want to study this model or how they model the situation that I present, it's very nice.

And one of the nonlinear equation that appears or model the situation is the blue one.

Here you have a second order in time derivative, second order in space derivative.

You have this nonlinear term and these two last term and a strong damping term here and this final term.

So the constant that appear in the blue equation are all these constants.

But for us, the most important is the Reynolds number and the Knudsen number.

And I will explain in the next slide why we need to keep in mind these two constants.

So in the thesis of Pablo and also in this talk, we focus our analysis in the linearized version of this blue equation.

So obviously we take epsilon equal to zero to erase this term, the nonlinear term.

And you obtain this, the viscose or dissipative van Beinhardt in the equation one.

So this is obviously the only constant that we need to keep in mind is the Reynolds number and the Knudsen number.

But one property that I found in the literature is very, very surprising.

If you have this condition on the parameters, the Knudsen number and the Reynolds and Knudsen number, you obtain a chaotic semigroup in the sense of the Van A.

So I will not go into the detail of the Van A chaos and the detail of the proof.

This is very interesting for me because if you have an equation like a wave equation with these two extra terms and the constant that appears here,

if it satisfies this red condition, then you obtain a chaotic semigroup.

So this is a paper of Conejero, Lisama and Murillo about the chaotic dynamic for the van Beinhardt and Aiden equation.

So when I looked at this paper, Pablo talked with me about the study, some control problem in his thesis.

And I said, OK, we can study the control property for this equation.

So this is the objective of the thesis of Pablo and also of this talk is to study the controllability of the equation,

but with the control localizing the boundary in the left hand side of the boundary of the interval zero one.

You put the control.

So but you need to keep in mind the constant. This is the most important part.

We need to to always keep in mind this condition of the constant.

Because we don't we don't want to to to have a chaotic dynamic for the equation.