# Summer school Control of Evolution Equations

## Course Content

Marius Tucsnak (Bordeaux):
Controllability of PDE systems at the crossing of functional and complex analysis with PDEs

These lectures aim to introduce the participants to the controllability theory for systems governed by linear evolution partial differential equations. This is a field where functional analysis provides a general framework to define various controllability types for infinite dimensional systems and allows some natural generalizations of the controllability criteria known in a finite dimensional setting (such as the Hautus test). However, when dealing with a given PDE system, its precise structure (such as the behavior of high frequency eigenvalues and eigenvectors) is essential for the controllability properties of the systems. Therefore, the controllability of a given PDE system is often studied by a “case by case” approach, often combining PDE techniques with methods from complex analysis or non harmonic Fourier series. To illustrate the interactions of the above mentioned fields we present some recent advances on parabolic PDEs and on systems coming from population dynamics with age structuring.

Hans Zwart (Twente):
Infinite-dimensional systems, state- and frequency domain techniques

In this course we will study evolution equations described by partial differential equations. Many of these partial differential equations model physical behaviours, and hence it is only natural to take this into account. Therefore, we write our partial differential equation as a port-Hamiltonian system. The Hamiltonian often equals the energy of the system. For this class we identify those boundary conditions for which the (homogeneous) evolution equation is well-posed, i.e., possesses a unique solution for every initial condition with finite energy.

By controlling some of the boundary conditions and observing others, we obtain a boundary control system with observations. Like any linear, time-invariant system a port-Hamiltonian system possesses a transfer function. For linear control, we show also how stability can be proved using the transfer function, i.e., using frequency domain techniques. Summarising, we address the following main topics:

• Existence of solutions for linear, homogeneous and inhomogeneous port-Hamiltonian systems.
• Transfer functions.
• Stabilisation of port-Hamiltonian systems, time and frequency domain techniques, linear and non-linear control.

A more detailled course description can be found here: