All the organizers, Wang Ye and Qu Peng and Professor Zuo Zuo and Professor Wang Ye and

Professor Lei. It's my way to talk in this workshop.

Actually, I will also take this opportunity to show my thanks to Professor Li Daxian and

Guenthe Lohia Green and Enrico Zuo Zuo for their support, constant support.

This workshop also reminds me of my first visit to Erlang Bin. It was in 2009,

I think. Since then, I have had many collaborations with the school in Erlang Bin.

I have worked with Guenthe, with Martin Gugart and other colleagues in Erlang Bin in my research.

Hope to see you soon, either in China, in Shanghai or in Erlang Bin.

Now let's move to my talk. Today I'm going to talk about

feedback stabilization and inverse problem for a non-local transport equation.

I will give my talk in three parts. First, the introduction and then the results and finally,

some perspectives. First, the motivation. It comes from the supply chain model, supply chain

management of semiconductor manufacturing system. This system has a so-called highly

reentrant character. It comes from two facts. The first is the numbers of parts manufactured

per unit time at each process is in very high volume. The second is the

constructive, consecutive production steps, numbers in very large, like you see the typical

manufacturing factory or production line has more than 660 wafers in process in the same line.

There are more than 500 work steps in the whole production process.

The left one describes the main process in the chip production process.

You see the oxidation, you see the lithography,

the extraction and the strip cleaning and also iron implantation, chemical vapor deposition,

metal deposition and so on. You see on the right figure, these processes are repeated from time to

time. The whole cycle is very long. Typically, it's like six weeks. The process is very long.

So we should say that the control or the management of such production process is very important

and also very difficult for these chip factories. Here is the PDE model,

which was initiated by some professors in Arizona State University and some colleagues from Intel

Corporation. It was in 2006. They used these conservation laws to describe the supply chain

model. Actually, you can see this is indeed a transport equation. The unknown function

road describes the density and lambda is the velocity function. This function is typically

nonlinear and it depends on the quantity W, which is indeed the total mass. It's the integral of

rho at each time. So you can see it's a nonlocal transport equation. If you give the initial value

rho zero and you give the boundary condition as the inflow here, rho multiplied with the velocity,

it's the influx, which means the rate of products entering the factory or the production line.

And on the other hand, you can measure the output of this system, which is defined by rho one times

lambda. It is indeed the outflux. It is also the rate of products exiting the factory. So in

control problem, especially in the stabilization or the inverse problems, we will design a

control u in terms of the measurement y. So it's a feedback control. And typically lambda is a C1

and positive function and the special model is one over one plus s, which we will use later.

There are some control and inverse problems that have been studied in the last decades.

So the first two are the classical controllability problem, which we will not mention too much.

And we will focus on the feedback stabilization problem, which can be described as follows.

We are trying to design a control u, which depends on the value of the output or the measurement y.

Here, what f can be a linear function or a nonlinear function even can be functional.

And with this feedback control, we want to stabilize the system, the closed loop system,

so that the solution can converge to a desired equilibrium as time goes to infinity.

And in the second page, we can see some other control problems which are more related to

applications. First is about the controllability on the nodar profile, which means we want to

drive the solution so that the outflux can reach some desired value, yd. Or we can consider

a demand tracking or backlog problems. The control target is different, and still they are very

important in practice. And the last one is the velocity recover problem, which we will