Okay, so welcome everyone. We have Professor Tia Lizzi from the University of Paris-Dauphin

and today he will be speaking about non-contrability results for the half heat equation on the

low line based on the called the Ferro Dal wave function and applications of question

equations. We are looking forward to you also, Dizzy.

Okay, thank you very much for the introduction and also thank you very much Marius and Enrique

for giving me the opportunity to present my work here in this seminar. So as you told,

the main subject of this talk will be two results of non-contrability and so in fact

the results in themselves are not very impressive but what is interesting here is rather the method

I used which is quite original and I hope that it can be maybe extended to many other situations.

Okay, so this is the outline of the talk. So first of all I will give some general facts

about linear control theory to be sure that everybody has the case to understand the rest

of the talk and then I will define and give some properties of the main tool I will use here,

meaning what I call the Prolique Solidar wave function. Then I will present the non-contrability

result on the half heat equation and to finish I will explain how to use this previous result

on the half heat equation on the Grushin equation and give a conclusion. Okay, so

first of all maybe I would like to explain a little bit what is the main, what is in fact

the main goals of control theory. So the general goal is to act on dynamical system, so you take

some dynamical system y prime is equal to f of y u, so y is the state of a physical or biological

system and u is a parameter that you can choose which is called the control and you can choose it

in order to ensure a certain number of properties. So roughly speaking you can decompose the

problem into three subclasses. The first type of problems would be what I recall the controllability

problem, so meaning that you take an initial condition in some class in some state space

and you would like to find a control such that at some given time capital T you can reach a

prescribed target set. So it is what we call an open loop system meaning that the control will

depend only on the initial condition and not at the evolution of the state over time. So this is

rather a theoretical question and it has many links notably with inverse problems or unique

continuation properties. A second issue is the question of stabilization, so in this case you

would like, so you take a typical system which is unstable and you would like to make it stable

by choosing a control which depends itself on the state y and you would like to ensure that at

infinite time for example you go to an equilibrium zero for example, so it has many implications in

robotics, electronics, fluid mechanics. And the last issue would be optimal control, so in this case

your goal will be to find some control use such that you minimize some cost functional depending

on y and u. So in this case contrary to the controllability problem you cannot really reach

some target state but your goal is to minimize some cost functionals. And here what I will focus

on in this talk is the first problem, so the problem of controllability.

Okay so the general framework is the following, so you take some linear operators for example

on Hilbert spaces you take and you consider a dynamical system of the form y prime is equal

to a1 plus bu, so y is the state of the system, b is what is called the control operator and you

have some parameter u which is the control, so this is the parameter that you can choose

in order to reach some goal as we will see afterwards. So the situation

so you can think about is the following, so you take a to be the Dirichlet Laplace operator

and bounded domain capital omega of rd for example and capital b will be for example the

the characteristic function of subsets little omega, so this is called distributed control,

so you have a control which is localized localized somewhere and you would like to act on the whole

whole domain thanks to this control localized on little omega. So here we will be interested in

what we call the the null controllability property, so what does it mean? It means that

for any initial condition in a state space you would like to find a control such that

you reach the equilibrium capital T, the equilibrium zero at time capital T and so of course

as soon as you have reached this equilibrium zero you can switch off the control and you will stay

at zero forever. So here it is a problem of existence, it means you have to find a control.