So I hope you had a nice lunch break.

So let's go on with the interpretation

of this slide with the description of the Lagrange

Hamilton approach for optimal control.

Now, so the left side is the compressed description

of what's going on there.

And the right side is the explanation

why this is the case.

So you take as a basis the Lagrange multiplier method.

And if you put together at all time points

the different dynamical equations here

as an addition to what you have to solve in the optimization

task there, then you would have many adjoint variables.

So for every time point, a new adjoint variable is there.

And the step from here to here is only

a rewriting of the first part of it.

So that here you have all the things together

which we have defined as a Lagrange function,

as a Hamilton function then.

And now here you have a sum from t equal to 0

to t equal to t plus 1.

This in the first place here is a lot of Hamilton functions.

And if I now start my summing only with t equal to 1,

then obviously I'm missing the part here

where you have to equal to 0.

And so this is something we have to take out of the sum here.

So I have Hamilton function from s0, u0, and lambda 1.

And then the next point is that in the original description,

the sum starts for lambda s.

It starts with lambda multiplied by s with the index 1.

Because if I fill in the 0 here, it's starting with 1.

And it's ending up with, because the index here has a shift,

it's ending up with t lambda transpose capital T

multiplied by s capital T. And now in this description

downside here, we have all the lambda s in it.

And the first one here is again the lambda 1 s1.

But here now the last one is not lambda t st

because if I fill in the t minus 1 here,

I have no time shift which is shifting it up to the capital T.

And so therefore the last element lambda t st

is outside of the sum here.

And if you take this description here and now play the game,

let's do a derivative down to st, ut, and lambda t.

What's going on then?

Now if I take the derivative down to the Lagrange parameters here,

to the joint parameters, or it is named in this condition here,

then you would have the dynamic equations again.

Second, if you take the first derivatives down to the s,

where is the best point to do it?

I think the best point to do it is here.