Thank you. Thank you very much. Thank you for the invitation. I'm very happy to see you again.

So as the title says, I will talk about a geomagnetic exact beams and very shortly I

will explain what I mean by that. I will talk about the case of one beam but also networks of

beams attached to each other and their tips. And then I will talk about well-poseness and

stabilisation problems and I think I will skip the controllability problems today.

So before starting just one thing about the notation, I want to introduce the following

notation, the HAT notation, which is just another way of writing the cross-projects.

And you will see this notation everywhere in the slide that follows. So the HAT notation is just

when you have two vectors, for instance u and zeta here on the slides in R3, we write the

cross-product between these two vectors also in the following way here in red as u hat applied to zeta.

So then u hat is just this matrix, this skew symmetric matrix on the screen.

And another operator is vec which is just the inverse of that. So when you apply vec

to u hat, you recover the u. So everywhere in the presentation the HAT is this operator.

So why do I speak about beams that are geometrically exact? That's because I'm

interested in beams which are very, very flexible. And when a beam is very flexible,

it will have large motions. And these large motions cannot be neglected. And if you want

to represent such a beam using a linear beam model, for instance the Euler-Bernoulli or

D'Himoschenko system, these motions will not describe accurately because the large motion

will be neglected with this model. So you need a non-linear beam model which is called geometrically

exact. So this is the kind of model I consider here. And in the governing system,

to account for this large motion, you will have additional non-linearities.

And the beam at time t, the position of the beam at time t, is given in terms of two states

which are p and r, and red and blue here. p is just the position of the reference line of the

beam, which is the red dashed line on the picture. And r is a rotation matrix whose columns give the

orientation of the cross sections, so the blue circles on the picture. So p is just a vector in

r3 and r matrix. So what about the mathematical model for these beams? I will speak about

two perspectives, two frameworks to represent the beam. The first one, which is most common,

is where you take as a nonce this p and r I just talked about. Then you get this system

in the middle of the slide, which is complicated. It looks a bit like a wave equation. You see that

it is second order in time and second order in space. Here the functions v and z, they depend

on the state. They are defined here at the end. It is quasi-linear. And you see also

in the equations and the definition of z, the appearance of the matrices m and c and the vector

epsilon c, these are just given data. They depend on the material and the geometry of the beam.

So this is one way to describe the beam with a complicated governing system with six equations.

But there is another way to look at it. It's when instead of taking p and r as the unknowns,

you take a v and z instead. So v and z, they have a specific meaning. So in v you have the linear

velocities, the angular velocities, two vectors in R3. In z, similarly, you have the internal forces,

internal moments. And so you can also take as a state y, which is made of v and z. Then y belongs

to R12. But the system is much nicer in the sense that now it's a first order in time,

first order in space. And most importantly, it's only semi-linear. So on the right hand side,

you have a quadratic nonlinearity. You can also write the system in a more compact way.

As here, you just need to multiply everything with the inverse of the

matrix on the left. And in addition to these properties, the matrix A is hyperbolic,

which implies that you can apply a change of variable and diagonalize your system.

So this is the second way of looking at the geometry of the exact beam. So you have these

two frameworks. The first one, which I just called JEP. And the second one, which I just called

IJEP, where I stands for intrinsic, because the variable y is often called intrinsic in the sense

that it doesn't depend on P and R. And an important observation here is that these two

frameworks, just by definition of y, they are related by a nonlinear transformation.

I call it T here. It's defined in the middle. And I will go back to this transformation later