I'm going to talk about some recent results about the controllability of the heat equation.

Indeed, the heat equation is an equation which is relatively simple compared with other parabolic

equations, but there is still, but however there is some research to do. There are still many open

questions. In particular, I will briefly explain three of them. The first one is to see how many

domains can we prove the null controllability. The second one is to, they are the transport

diffusion equations, and the third one are the average controllability properties. So the first

one which corresponds to one of my papers is the null controllability of the heat equation,

you see, which is the larger family of domains to which we can prove the null controllability.

Just in case I recall you that the null controllability problem is having some initial

value y0 to see if with the help of a control we can manage that in some time t the solution is

0 if we can take any solutions to 0 and in addition and if we can do it continuously with

respect to the initial value. So that's the statement of the problem. Also I recall you

just in case that an equivalent problem is the observability problem to see for any free solution

of the heat equation knowing the size of the function in the observation domain. If we can

have an estimation of the size of the function at time t. Both problems are equivalent, the

controllability problem and observability problem. This constant is even the same one by the Hilbert

uniqueness method. So I usually mention both indistinctively as they are equivalent. Okay,

first of all I will do a small recap of the historical approaches that have been used. The

first one are transformations involving the wave equation that usually are to multiply by some kernel,

get the wave equation and use no facts of the wave equation to get other facts out of the heat

equation. It has the advantage that is really precise but it has the disadvantage that it does

not work in any domain, in every domain as the wave equation must be controllable so that we can

apply the method which is not always the case. The second one are spectral inequalities. They are

also known as the Leboard-Rovien technique that consists of controlling some frequencies, then

letting others to decay, controlling another frequencies, letting the rest to decay, etc.

up to a geometric sum. And the third one are Karlemann estimates which consist basically on

multiplying by an exponential wave and doing integration by parts. So these are the main three

approaches. What success have they had? Well, Leboard-Rovien approved in 1995 the new controllability

of the heat equation in many analytic domains. Afterwards, Furzykow and Imanowilow, well,

almost simultaneously Furzykow and Imanowilow, proved the same result for C2 domains. They

used the Karlemann inequalities. Afterwards, in 2013, Liu adapted the Leboard-Rovien approved

proof for C2 domains. And in 2014, Apraiz, Uskaria, Tsav, Wang and Zhang proved that the

heat equation is not controllable in all locally starship domains, which is a family that is bigger

than C2 domains, but still is not all the Leipzig domains, which should be the main objective.

So what I proposed was to see if the technique of Furzykow-Imanowilow, the Karlemann inequalities,

could be extended to all Leipzig domains. In particular, I focused on pseudo cylinders,

which as you can see in the picture, contain domains that contain a cylindrical part and that

are bounded by some Leipzig graph. These domains, well, I interested in those domains because of

two reasons. The first one is that they appear in fluid mechanics. This, well, you have to shift

the cylinder, but this part can represent the floor of a river or of the sea, especially in

dimension three. So yes, so they are, these domains do appear in fluid mechanics. And the

second reason is that a Leipzig domain can be split into some pseudo cylinders. You can

split the band down into some pseudo cylinders and then an interior regular domain. So for me,

proving pseudo cylinders was the first step toward proving the new controllability of the

heat equation in any Leipzig domain. So what I did prove was that it is indeed

null controllable with the Karlemann inequality and that we get the usual, that the cost of the

control was the usual one. So what were the difficulties I have to face with respect to

full psychopharmacological work? The main one was that there is an auxiliary function in the

weight and there was an auxiliary function which does not exist in Leipzig domain. The main reason

are corners. The domain has corners. So these four conditions are that it is to be fit to, to be null on