Professor Martin Lassar, he's a professor at the University of Dubrovnik in Croatia,

and he's a visiting scientist, I think until still half a year, right?

Like until, yeah.

So there's more time to follow up questions and hopefully collaborations.

And he will give a talk today on the topic of optimal control of parabolic equations,

a spectral calculus based approach.

Okay, thank you Tobias.

Thank you all for joining this workshop, for joining us.

Also thanks to the chair of dynamics, control and numerics here at Fouling University in

Erlangen and its host professors as well for organizing this workshop and letting us opportunity

to present our recent results.

So today I'm going to present some optimal controllability results.

I will start immediately with the problem we deal with.

So we consider optimal control problem for an abstract parabolic equation of this form,

whose dynamics is governed by operator A, self-adjoint, in general unbounded operator

H, A acting on Hilbert space H, which in general is assumed to be infinite dimensional.

By S, T, we denote the semi-group generated by this operator.

The keynote example that we can use for this kind of setting is of course the Dirichlet-Laplacian

as well as accompanied by some suitable boundary conditions.

The control enters the system here through the initial datum and its aim is twofold.

One aim is to minimize some given energy functional in such a way we obtain an optimization problem,

but we accompany it with an additional constraint and that is to hit some prescribed target

as close as possible.

So let me just, now of course such kind of problems can be also considered as type of

inverse problems of source identification and due to the dissipativity of parabolic

problems, especially of the heat equation, we know that such problems are numerically

very challenging.

And now let us just briefly recall some basic notion and facts about the controllability

of such systems.

So as usually we say system is controllable to some target state y star in given time

framework capital T if there exists control that steers the system to the target.

So in our framework the system we consider is not controllable in general.

Again we can take as an example Dirichlet-Laplacian for which we know that its image is contained

in the domain of the operator in such way if you have non-smooth target you cannot reach

it by any control in any time.

But on the other side due to the Hahn-Bannach theorem and the properties of the operator

A we know that its image is dense in age.

In such a way we have the approximate controllability property for this system which means that

any target can be reached within some upper given precision epsilon.

Of course if the system is approximately controllable if you are given some target and epsilon then

there are many controls that can steer your system to the target ball.

Actually infinite many of them.

And then it makes sense somehow to select the optimal one.

In order to choose the optimal one we have to specify some optimization criteria and

that consists of minimizing a suitable cost functional.

And here I can formulate the precise problem we are dealing with.

So the problem is of minimizing the cost functional at the same time steering the system within

the epsilon distance from the target y star.

The functional consists as usual of the cost of the control but also of the tracking term

which penalizes the deviation of the system of the trajectory of the system from some