So Professor Mike Consonego is a current research guest at the FAU Erlang in Nuremberg, and we

are very happy that he is here.

He is from Brazil and from the University of Itaúba, M.G.

And today he will give a talk about control of heterogeneous 1D reaction diffusion equations.

Thank you very much.

Okay, thank you.

I'm very happy to be here, talk to you and being part of this group.

I'm a DAAD fellow and what I'm going to talk about now is part of the project I'm developing with Professor Henrique.

I take this opportunity to thank Professor Henrique for all the attention you have devoted to me.

I have to say that this is a very recent study and what I have are just a few partial results.

So let's go ahead.

So this is the problem, a parabolic problem with heteros, heteros terms A and B.

A control-led reaction diffusion equation with this heterogeneous terms A and B with controls U and V acting at zero and L.

So here we consider F a monostable function or bistable function.

For instance, this function or this function with theta here and the A and B we consider positive functions of the last two.

And the controls U and V are measurable functions satisfying these constraints.

What we want to hear is to control the problem in the following sense.

I choose a target, it can be homogeneous target or heterogeneous target.

And we want to put the controls here and here to drive the trajectories to this target.

So here we have the definition of this.

Let Z be a static solution of one that sees a solution independent of T, a solution of the stationary solution between zero and one.

And we say that the control-led equation is controllable infinite time towards Z if you have this.

And controllable infinite time towards Z if you have this.

So in fact, I'm only going to talk about the results I got for this situation.

Infinite time towards that state Z.

Results with finite time control are not yet complete, but I believe that they must also occur.

So here just this situation I will consider, I will deal.

Here we have some recent reference about this topic.

And in these words, we can see many phenomena that such problems cover.

For example, we can see the population dynamics, chemical reactions, magnetic systems, linguistics.

And the first, it's the same problem like that, but with A equals B equals one.

That is a homogeneous problem.

And with phase portrait, they got a lot of results of control towards some homogeneous state states.

The second, they got a lot of results for multidimensional domains, in short, but in three, they deal with some problems with heterogeneity terms.

And I want to say that none of these works addresses a case as general as this, as that I want to deal here.

And this was my main motivation.

It's well known that with heterogeneous terms, the problem becomes more realistic.

So this was my main motivation to study this problem.

Okay.

The main result, our strategy will be to use the Matano results on the asymptotic behavior of solutions of similar problems.

In short, this result of doing Matano says that the solutions of a parabolic problem converge to a state state solution when D equals two infinity.

Obviously, this is not always true, but in this case, this.

So the key argument will be to find conditions to have uniqueness of the statuinary problem associated with the problem one.

So, because if you have a unique stat state solution, all using this result, all trajectories will converge towards it.

And this is what I want to do, essentially is to find conditions for to have a unique solution, a unique stat state solution for the parabolic problem.

Okay, now I need some technical results and to initiate my main result.

And I'm going to run a little here with this a little technical.

First we present a very general result of as this and uniqueness.

So let G assume function such that where G1 is like this and G2 like this.

And this inequality is related to Lipschitz condition for that results about the uniqueness and the extent in all the theory.