So welcome everybody. We have today Professor Yacine Chitou from the University of Paris

Sackler and he will be speaking about asymptotic behaviors of one-dimensional weight equation

which sets valid boundary damping. Please Professor Chitou, we're looking forward to

your talk.

Okay, so hello again everybody. Thank you very much for this invitation and I will speak

for around maybe 50 minutes about the topic, a very well known and very much studied topic

of stabilization, boundary stabilization of a one-dimensional wave equation and as it

is written in this talk, this is a joint work with the Swan Marks and Guillermin Mazanti

who do me the pleasure and the honor today to be with us. Okay, so look in the title,

there is this set valued boundary damping. Already for those who are aware of this topic,

they should not consider the set value, the point of view as an exotic thing, but it will

be clear hopefully from the talk. Okay, let us start. This is the plane of the talk. There

will be a rather long introduction to recall previous works and because really this is

a well-studied subject. Many things have been done and it is important to recall and to

recall all the main issues and the main results that have been opened. Then the second part

that will explain our point of view, why set valued and then we will address the standard

issues of well-positiveness asymptotic behavior and I will finish with something more common

in control theory, this ISS concept, the stability concept called input to state stability if

I have enough time. Okay, let us start now with the introduction and here is the PDE.

So it's again, I insist on it is a one-dimensional wave equation on the segment zero one with

a boundary control or boundary stabilization in the sense that there is a relationship

between the partial derivative. So at x equal to one for all time, there is this relationship

between the partial derivative with respect to x and the partial derivative with respect

to t and usually this relationship takes the form that is written there, namely partial

derivative of x is a function of sigma minus sigma of the partial derivative of t. Of course,

this point of view can be understood as a localized boundary control. Imagine that the

right hand side of this equation is the control u that you choose as a feedback function of

the partial derivative with respect to the time. Why? Sigma does not need to be linear.

So in order to modelize a non-linear phenomena, okay several non-linearities in components

and there have been many works where the sigma takes the form of a saturation function like

the arc tangent function. And so there is a list of many works that we consider as important

in this setting where people address the issues of stabilization and mainly asymptotic behaviors

depending on the shape of the function sigma. Then you can assert, you can study the asymptotic

behaviors. Not also that many of these works address these issues in higher dimension,

higher space dimension here in dimension one, namely I recall that x belongs to the segment

zero one, but many of these works have considered domains, special domains in two or three dimension.

Okay, in this list of work let me recall a few of them, especially one of Van Kostel

Noblon Martinez in 2000 where they considered cases where the function sigma is not increasing.

And I will recall also the reference of Alabao of 2012 which is actually a survey where she

recalls all the previous works and all the asymptotic behaviors that have been achieved

and proved in the literature before 2012. Finally, let me point out the last reference

where sigma is the sine function and the authors had to take care of how to define correctly

the or precisely the problem because sigma was not the first time that sigma was a set

valued function. Okay, now what are the questions that people looked at? So first of all, existence

and our uniqueness of solutions. Okay, so I recall in the top right the dynamics of

the problem. So the first issue was existence and uniqueness and once this is proved, people

had addressed the issue of the asymptotic behavior, namely usually under very reasonable

assumptions on sigma trajectories tend to zero in the appropriate norm and the asymptotic

behavior issue asks to precise this decay to zero. There is a classical functional framework,

namely the space x2 which is in the variable z this h1 star which is h1 with a boundary