Thank you, Enrique, for this kind introduction.

Yeah, that's precisely the summary of what I'm going to talk about.

I would like to talk about challenging optimal control problems from practice, from academia,

and how one can solve them even when they are very large.

And I would like to spend the first half of my talk on modern predictive control and the

second half of my talk on reinforcement learning.

This is material that has evolved over the last 10, 15 years, so there are a lot of coworkers.

I would like to point out two coworkers for this use case, which is a real industrial

use case, which is by Matthias Höger, who is a former student of mine, and Kilian Link,

who are both at Siemens Energy here in Erlang and just 500 meters away.

Yeah, so let me start with a brief introduction.

What is this all about?

This is about optimal control.

So we need some control system, and let me start right with a formula.

So this is a control system here in continuous time, where we have a differential equation.

This is a control system in discrete time, where G defines the next state at a discrete

time instant.

And for this talk, it doesn't really matter.

It will apply to both, more or less.

I will, of course, not do everything for both models, but I will switch from one to the

other occasionally, whatever is more convenient.

Okay, so X is the state.

This is what we want to control.

U is the control input.

That's the variable time depending that we are allowed to change according to our needs.

And many control problems, particularly if they last longer than just a few seconds or

whatever the dynamic scale is, will require optimal control in feedback form.

What's a feedback control?

A feedback control means that we do not pre-specify this control U of t as a variable or as a

function of t for a long time into the future, but actually would like to have our control

as a map F that takes the state or a measurement of the state and produces the control value

from that.

And the main feedback is completely clear.

We look at what the system is doing, and this feedback from the system tells us how to control

it in the new.

Yeah, the reason for that is that with the feedback, we are much more able to correct

deviations and deviations are everywhere.

We have modeling errors.

We have perturbations that we have not foreseen.

And just to show you a little quick example, it's this one.

This is a pendulum example that a student of mine did a while ago.

So it's a swing up of the pendulum.

This, of course, you could do with open loop, but then the feedback is needed because the

student is going to tease the pendulum and the tensor realizes that the pendulum is pushed

and reacts to that by swinging it up again if necessary and making correct.

That's the mechanism of feedback that is what provides the robustness that we want to have

in most practical applications.

Okay.

Now, I would like to talk about control problems on infinite horizon today.

There are many interesting control problems that are on finite horizons, but I would like