We are dealing with the problem of aeronautic optimal shape design.

So in aeronautic optimal shape design, we are dealing with an important topic for guaranteeing

global transportation of persons and goods, and on the other hand, highly technological,

in which not only efficiency but also safety is very important for all of us.

And as we said, this is a complex geometrical problem because after all, the object we have

to optimize is not just one or two or three parameters, right, it's a complicated, say,

three-dimensional domain, right, which is the cavity of the airplane within the space R3.

So you see, something that is interesting is that, you know, it's obvious in some sense, but

it immediately shows one of the ingredients of the complexity of optimal design problems

is that, of course, we are considering shapes, right, which are open bounded domains

of R3, these are the shapes under consideration. Of course, the dimension of R3 is equal to three,

but if we consider, you know, the space S, the space of shapes,

what is the dimension of that space?

Could you please tell me what is the dimension of the space of shapes in three space dimensions?

Does anyone have any hint about this?

Some comments, I think.

Infinity, precisely. This is something very important, right? So we are living in R3,

the shapes we are considering in R3, but the space of shapes is itself infinite dimensional, right?

And you can see that very easily when you are looking to, you know, spheres, right? If you look

only to the spheres, assume we were just designing balloons, spheres, what would be the dimension of

the space of the spheres?

If we just consider spheres, the dimension could be, would be rather limited, right? Because

basically it will be three dimensions for the center of gravity of the sphere,

for the center of the sphere, and one more dimension for the radius of the sphere. So it

will be dimension four, right? But of course, I mean, you can go to more and more complicated

polygons and polyhedra, right? Which will be needed in order to be able to approximate any shape.

And you see how as soon as you are increasing the number of sides, right, in this polygonal

or this polyhedron, the dimension is growing endless, right? So even if we are considering

a simple geometrical problem in R3, the mathematical problem of optimizing the shape, right,

is formulated in a space of infinite dimensions, right? Where you encounter all the difficulties

related in particular to the fact that bounded sequences do not necessarily converge, right?

They will only weakly converge in the case of reflexive Banach spaces, and this will generate,

of course, many technical difficulties. So this on one hand, the classical problems in industry

in which you try to optimize shapes in order to generate a kind of optimal process,

these problems are infinite dimensional. Now, they are infinite dimensional problems in the sense that

you are considering parameters, the variables to be optimized, live in an infinite dimensional domain.

On the other hand, the criterion you have to optimize in order to design the airplane

are multi-fold. They are multi-fold, right? So in particular, in order to analyze the stability

of flight, you need to consider the drag, the thrust, the lift, and the gravity, right? So

the function that you will have to minimize as you do in multi-criteria optimization problem, right?

So if you do multi-criteria optimization, the way we do it is we say, well, the functional J

that I'm going to minimize will be a weighted combination of several functionals, right?

So one functional will take into account the lift properties of the airplane, the other one,

the drag, the other one, the thrust. There could be criteria also related to the weight of the aircraft.

There could be criteria about the volume, the height, the comfort of passengers inside,

the oil consumption. There are many, many different criteria you need to optimize, right? So and then

you know, the question will be how many of these criteria you put into the functional you are

minimizing? That's first. And the second one is how do you weigh the different coefficients, right?

Because of course, depending on how do you choose these weights, alpha i, you are giving more

relevance to one functional than to the other. Okay, so you can see that when you are simply