It is for me a great pleasure to introduce you all Professor Qi Liu from Sichuan University

in Chengdu. He is a well-known expert in PD control, in particular with emphasis on stochastic

models. I know now Qi for many years since he was a young student. He has developed since

then a brilliant career. He was in particular, among many other awards and recognitions,

he was invited as a speaker at the International Conference of Mathematicians last summer.

So Qi was a PhD student of Xu Zhang. Xu Zhang was a PhD student of I think,Yu Min Yong.

I think Yu Min Yong was a PhD student of Professor Li Tatian. So we are back to the roots. So

we see here, I think the fourth generation of descendants of this very noble and fruitful

school that Professor Li Tatian in Fudan has developed. I remember that I met Xu Zhang

in I think it was in Hanzhou. He was finishing his PhD. That was probably like in year 2000

or maybe 99. I was for a conference there. And at that time I was the PhD advisor of

a student Antonio Lopez in Complutense de Madrid. And we had some results on the limit,

singular limit at the controllability level from waves to heat equation. We were able

to do it in one space dimension, but in multiple space dimensions we were missing some uniform

observability estimate. And it was Xu who did that rather quickly and efficiently using

point-wise Karleman estimates. Since then I have been collaborating with Xu for over

20 years. And I also have the honor and the pleasure of visiting Sichuan University very

often, actually almost every summer until unfortunately 2019. And I think I hope we

can recover these mutual visits and exchanges soon. She was also a postdoc in our team in

Bilbao for one year. She was also a postdoc for one year in Paris with Jean-Michel Coron.

But I think the best is to let him talk and to tell us about his novel findings on PD

control for stochastic models. So the stage is yours.

Thank you, Eric, for the nice introduction and thanks to the organizers for inviting

me to give a talk here. I'm very proud of this. And since the internet, now I'm in a

hotel, the internet is a lot of work. I shut down the camera. Okay. Today I will share

my screen. So where is the? Oh, okay. Here. So can you see that?

Yes. Yes, we see. Maybe you can put the panoramic

version. Okay. Now it's better. Thank you. Yeah. Okay. Yeah. Thank you. Yeah. Indeed,

the motivation to do this work was when I do my postdoc with Eric, he asked me what

about the exact control or wave equation. At that time, I get a negative result. I say,

oh, the stochastic version is not exactly controllable. Then I stop it. But then after

some time, when I read some papers from the physics, I realized the problem is the stochastic

wave phenomenon is not exactly controllable. But the problem is there is something wrong

with the model. That's the motivation of this paper. So thank you, Eric, for giving me,

providing me so nice questions. Okay. Since there are some professors that maybe work

on inverse problems, so I use some sentence to talk the controllability a little bit.

In my opinion, when we say we want to control something, that means we want to modify the

behavior of our system to achieve our desired goal. And if we can do this, we hope we can

do that in an optimal way. So this will lead to two natural questions. One is the controllability.

That means it may be our ability of the control. It shows the ability of the control. And another

one is the optimal control. Today, I will focus on controllability for stochastic wave

equations, one of the most typical stochastic models. But before that, I use a simple example

to explain what's the meaning of controllability for people from PDE or from inverse problems.

Consider the following controlled linear ODG. You say here, our B, B u is our control. What

we want to do is that for a given initial data y0 and for the desired destination y1,

we want to find the control u such that the solution to this system fulfills y capital

T equal to y1. That means we want to use the control to push the system, push the state

of the system to a desired destination. If a system is exactly controllable, then we

say that this system enjoys very good control properties. And then we can do many things,

like stabilization, like the noise. Yeah, we can try to decoupling the noise and the