And today I would just some several control problems on the network

hyperbolic systems and keeps a model and efficient simulation in the picture. So

let's begin by understanding what our networked hyperbolic systems are. So as

we all know that hyperbolic PDEs are class of PDEs that describe the wave propagation or other

phenomenons where the information can be transferred by a finite speed and such as the

traffic flow or the fluid dynamics and the mechanical vibrations you have seen in the

previous talk. And when these systems are interconnected in a network by addressing

specific interface conditions at the nodes, they form a networked hyperbolic systems. And if you

look around of the world around you, you may find applications that are modeled by such kind of

networks. And at the right side I bring some projects now at a chair which related to this

topic and some mobility program that we are running with China. So at the left side I bring

some examples that everyone may familiar with. For example the mechanical vibrations. The first

picture shows the nonlinear vibrating strings and also in some applications for example the

flight devices or wind turbine. We may model the large motion by the highly flexible beams. And

also you can see that in the soft robotic arms. And the left two corner and I introduce two

types of the transport network. For example the gas flow modeled by the esoteric molar equations

to describe the gas flow transport as the pipe flow. And also you can see that the water flow

in how to say that in the horizontal scale transferred on open canals. And the similar

non-system is a well-known model for us. And motivated by all these applications there are

three main areas of interest that we are interested in. So first is the modeling and analysis. And the

modeling part can do can be done in two ways. First is the physical driven so that we need

a meaningful physical models which need a lot of physical knowledge and the mathematics. And another

example is the data driven. So now for example to use the machine learning techniques to build up

some surrogate model. And then upon this models done and we can do the analysis. And in the following

talk you will also see that a difficulty may arise in network case especially when we have

a non-linearity in the networks are happening in the elements also could happen in the coupling

at the interface. And of course when we deal with the network case we may consider the topological

structures which may lead to different the contraband property. So the second interest we

are working on is about the control theory and optimal design. So there are two branches. So I'm

working for many years on the contraband properties for a high body case that is to find at least one

way to reach a target. And the target of course can be set in different ways. You may know the

classic way is to drive the solution to at a given time t to a final state. And on the network you

can also shift the control target to a nodal control of the profile. So which means I only

need the trace of the solution at one node to satisfy the given demand. So another important

branch is the optimal control or yeah to find at least the best way or efficient way to drive

the system to the desired state widely. So today we I in the following slides I will bring some

control results I've done in this years with the professor Gwint-Lagree and Tat-Sien Lee for

different types of 1D hyperbolic networks and you will see the coupling and the nonlinear team

may happen in the modeling and analysis. So let's begin with a toy example that now we have the

vibrating strings modeled by a very simple wave equations. They are coupled at the node 0 and of

course at one end we suppose that they are collapsed so we adjust the deletion boundary

conditions and the other two end I adjust the applied as the Neumann controls and at the joint

node you have the classical transmission conditions. And many results can be found for this linear case

for example the book that from Degas and Zuozhuo in 2006 and they have already proved that you need

at least two controls to drive the solution from any initial data to a given data at a given time t

and because now we are doing about hyperbolic case so the control has to be transferred from

one end to everywhere in the domain. So that means that we need at least a waiting time to make sure

that the information can be transferred so we call it's a controllability time for the hyperbolic

case that is also a typical property will be discussed in many literatures. And in a second

clone we adjust another boundary control problem we call controllability of nodal profile as I just