So I was thanking the organizers for giving me the opportunity to present some results.

It turns out that I finished something quite recently, so this is the first time when I

present these new results which look quite challenging to me.

That's the outline of my talk. I will go directly to the introduction with the applied part of my talk.

It is this slide, an example of a time optimal control problem. You have a rocket engine,

a BMW for instance, and you want to go from point A to point B.

Your initial velocity is zero and your final velocity aims to be also zero.

So that's the purpose and the force at your disposal to accelerate or to use the brake is bounded.

So the force is between minus one and one typically.

So the question is how to do this in the shortest time possible.

The answer, which can be obtained by elementary but not obvious, completely obvious arguments,

is that you should accelerate as much as you can till the middle of your trajectory

and then brake as much as you can, use the brake as much as you can till the end.

So by symmetry clearly you will stop and this is the optimal policy, of course not for the comfort of the passenger,

but from the time viewpoint we have here a very elementary example of what is called the bank-bank control.

Here it is for a time optimal control problem, we can have the same things for the normal optimal control problem.

So I describe again these problems on a PDE example where I will also state some results which are new already.

The way I will prove it, it will be explained a bit after, it will be in an abstract setting,

but one of the applications can be this problem.

So imagine you have a room which is not necessarily rectangular, it can be either rectangular or a C2 boundary,

and you have the heat equation inside and you heat somewhere on the boundary, on an open subset of the boundary.

You have, on the rest of the boundary you keep the temperature constant, let's say, we can use other boundary conditions,

and we have some initial state. So that's the system I consider extremely elementary one,

and which are the norm and the time optimal controls I will look at.

Already here there is a point which has been rarely addressed in the literature for technical reasons essentially,

because it's the most natural one, it's to minimize, so to go to obtain a certain temperature profile,

or to drive, yes, to obtain a certain temperature profile with a control which has the smallest possible norm in L infinity in spacetime.

So these problems have been looked a lot with L infinity in time, L2 in space, much less with this kind of norms,

and whereas the norm optimal control problems want you to steer the temperature to zero by controls which are bounded again in the same norm,

so I have a point-wise constraint, and we want to do this in a time which is as short as possible.

So these are the statements of this problem on these very basic examples.

So this is the time optimal control problem.

I impose a bound on the L infinity norm in spacetime, and I want to find tau infinity of m,

which is the smallest time in which we can steer the system to rest, and the associated control function which is denoted here by U m infinity.

Okay, so why do we have that? So I already stated the results we have, which are new, which are in fact completely new.

I will do some references immediately on the next slide.

So here the norm optimal control problem has at least one solution, which is bang-bang.

So the control which does the job somehow with the minimal energy is of absolute value, which is constant, of modulus which is constant all the time,

and at every point where the control is applied.

But it is part of being a constant. In practice, it's something which is extremely oscillating between those values.

But this is something we don't discuss here.

And for the time optimal control problem, the situation is the same.

I have a solution, at least one solution, and this solution is bang-bang.

So this is a kind of strong bang-bang property because it's at every t, almost every t, and every x.

And moreover, these two functions are inverse, are reciprocal.

The functions n infinity and tau infinity. So n infinity is a norm optimal control function, tau infinity is a time optimal control function.

With these norms, the infinity comes from the norm.

So is the norm optimal in infinity?

No, point-wise. N infinity in spacetime.

So is norm optimal in infinity?