Welcome everyone again to this last seminar of the month in July.

And today we have Dr. Alexander Keimer from the UC Beckley University talking about the

vulnerability of non-local conservation laws on bounded domains.

So please you can use the chat for the questions during the talk and of course you would be

unmuted after the talk so that we can interact with the speaker if you would like to do so.

So please Alex you can start.

Thank you very much for the introduction and also thank you very much for attending the

talk.

So I should start with saying that this is joint work with Alex Bayer and with Jean-Michel

Coran and with Nicolas Dainity and Lucas Ploot and since I guess at least two people from

my co-workers are in the audience I also asked them to add let's say two details and then

I might miss some parts which I guess will happen for sure.

Yeah so I will talk about the vulnerability of non-local conservation laws and let me

give a short overview.

What I will first talk about is why do we care about these equations and then I will

give a problem statement and basic results on existence and uniqueness of solutions and

then we will tackle reachability, exact boundary and final state controllability and on-time

behavior and we conclude with steady state solutions and linearized or the linearized

system to the corresponding non-linear system.

Yeah so why non-local and what does it mean?

So what you can see is basically a sci-fi broadcast from this year and some of you might

know this and so this is Star Trek Picard but what's interesting is that the term non-local

even has made it into sci-fi so from that we can hope to use that it becomes more and

more important and that is one reason why we study this.

Other reasons are that non-local conservation laws or more general also non-local dynamics

are used for modeling many many different aspects.

Here we see one which was more or less the first I guess as far as I know the first application

which is here for supply chains so just to introduce you to the notation and to the equation

so you can see on the left hand side conservation law and you can see the flux function lambda

or not the flux function but here the velocity function lambda and so you can see it's a

transport equation and some of the transporting velocity depends on the average load on a

given edge so basically we're not given link so you'll be assuming that the link is parametrized

by zero one and you're interested in changing let's say the velocity based on this average

chain so why is it non-local because you integrate your density from over the entire domain here

so zero one and this is clearly a non-local effect.

We have additional initial datum and in this specific case you assume that your velocity

function is positive so that means that you can also pre-stripe left hand side boundary

datum and then you can look into existence uniqueness, optimal control etc.

This has been done quite a few years ago.

So then another application is traffic flow.

Here I just was writing down a simplified equation so again we model traffic flow via

macroscopically so that means you have a density and Q which is space and time dependent and

here we assume that the density is normalized between zero and one so one meaning high density

or basically maximum zero meaning the road is empty and again you have here non-linear

conservation law again and you see on the right hand side this one minus W term which

is representing the velocity and this is multiplied with the density so when you would go back

to the usual and first order conservation law models and then this velocity would usually

be something like when it's creature velocity one minus Q right but here what we do is we

replace this one minus Q by a one minus W term and below you can see how the W term

is defined so basically the W term is defined as an integral from X to X plus eta over the