Start today's session there will be two workshops today by PhD students and postdocs here at the

University. We can start our session at 2.15 in the afternoon there will be the second part of

this workshop and you Francois you will also have 15 minutes if you want to complete part of your

seminar of yesterday or if you want to intervene in some topics that are presented today in the

workshop so we can start and the first speaker is Mikael Schuster who is postdoc here at the

so thank you Julia I hope everyone is could hear me thank you for this introduction yeah I'm a

postdoc now for quite a few years here in Eilangen and my talk today will be about the turnpike

phenomenon for optimal control problems and uncertainty so this is ongoing work so recent

results and current work together with a colleague from Nansan University in Nagoya called

Professor Sakamoto so I first want to start with a short introduction about the turnpike phenomenon

maybe not everyone is aware what exactly is the turnpike phenomenon and this came up in economics

in 1958 so more than 50 years ago basically there is a highway in Florida in United States

called the turnpike the turnpike highway it's a connection between Miami and Orlando city of

and the idea is that when you are close to the highway when you're living close to the highway

and you want to travel between the two destinations it will always pace off to really use the highway

but if you are far away from the highway then maybe there will be a faster way which don't

touch the highway so maybe if you're living here somewhere close to the seaside then maybe you will

not use the blue turnpike but if you are living somewhere close to the turnpike and your destination

somewhere close to the turnpike then it will always pay off to use the turnpike and when we

translate this to mathematics basically to control theory for example that means if we consider some

dynamical system some time-dependent optimal control or control problem then there will always

be a connection to some corresponding turnpike to some steady state in the sense that if you

consider a long time horizon then the dynamic so the time-dependent optimal control will be

closed to the steady state control or to the corresponding turnpike for quite a long time and

this is what we're gonna analyze in the next maybe 15 to 20 minutes so as a motivation we consider

an optimal control problem with time varying feedback control given by some objective by some

integral objective depending on the control and depending on the state we consider let's say a

simple linear transport equation with some initial condition and some boundary feedback control in

the sense that the control given at the beginning of the space interval depends on the state at the

end of the space interval you can imagine for example that possible application is traffic so

transport of vehicles on a street maybe you will control the traffic light at the beginning of the

street but this should depend on the amount of cars in the middle or at the end of the street

or for example another application would be the distribution of some pollutants in a water channel

or water pipeline there you want to think about okay what do I need to insert in the pipe in the

beginning to guarantee that the pollution in the end is not so high by measuring the current state

of the pollution in the end so this is maybe a motivation why we consider this kind of transport

equation with the time varying feedback control and if we start very simple just doing some

simulation without doing mathematical analysis the first thing we observe is that so the application

here was we just multiply the control with the final state so there was no G around there's just

a multiplication then we exactly see this kind of turnpike behavior in the sense that we start

from some initial state then the control is quite active trying to steer the optimal state to the

turnpike which is the red line in the background then we stay at the turnpike for quite a long

time and in the very end we leave the turnpike to reach the let's say final destination in case

of the transport equation here the final destination is not really a final destination this bump or the

sink in the end is just because when we start the control let's say here or later than here then the

control won't reach the end so in the optimal control problem the control in the end is actually

zero because it will just not reach the end of the interval so and then a very important question

for us is what if the system is not deterministic what if some uncertainty appears in the system I

mean many applications are modeled as deterministic systems but actually there is a lot of uncertainty

in the system by parameters are not clear as I said for example in the pipeline water pipeline