We are pleased to present the DISC Winter Course 2020 on "Numerics and Control"

The course on Numerics and Control will deal with distributed parameter systems with boundary control and observation, described as port-Hamiltonian systems (pHs). The pHs representation emphasizes the power flows between subsystems and over the system boundary, and is therefore particularly favourable for multi-physics modelling and for energy-based control designs.

The focus in this course will be structure-preserving numerical methods, which transform an infinite-dimensional pHs into a finite-dimensional one, and/or a continuous-time pHs into a discrete-time pHs, mimicking the structural power balance at the discrete level.

We introduce specific numerical methods which achieve this goal, such as the Partitioned Finite Element Method (PFEM). The mathematical language will be vector calculus at the beginning, and exterior differential calculus at the end. Examples will be first treated thoroughly in one space dimension, and in a second stage in higher space dimension. The last part of the course is devoted to geometric time integration, discrete-time pHs and the use of the introduced concepts for discrete-time energy-based control.

In order to be as concrete as possible, two labs are planned for structure-preserving PFEM: the first one in Matlab will enable to tackle 1D problems quite easily, whereas the second one in Python will pave the way for simulation and control of 2D and 3D problems, making use of FEniCS software. The part on discrete-time systems and control will be accompanied by Matlab examples.

**1) Introduction to port-Hamiltonian systems (pHs)**

The pHs framework is presented for physically-based control of dynamical systems. It is rst recalled for lumped parameter systems, and especially the linear case is fully detailed, the link with the classical state space description is given (A = JQ, B, C = B0Q). The notion of Dirac structure is introduced. The extension to distributed parameter systems is provided through examples, essentially the wave equation in 1D. The notion of Stokes-Dirac structure is presented. The goal of structure-preserving numerical methods can then be clearly defined, and will be the core of the parts 2, 3 and 4. Moreover, in order to remain self-contained, an introduction to the Finite Element Method (FEM) will be provided at this stage on the special case of systems in one space variable: Lagrange and Hermite polynomial bases.

**2) Boundary control of PDEs in one space dimension**

This part will provide theory, numerics through the Partitioned Finite Element Method (PFEM) and worked-out examples on the following models: waves, Euler-Bernoulli beam and heat equation. The main advantage in the 1D case is that the controls / observations at the boundary are of nite dimension, so it is quite easy and natural to understand and follow the methodoloy with no extra functional analysis complexity.

**3) Boundary control of PDEs in higher space dimension**

Here comes an extra difficulty : the boundary controls / observation live in an infinite-dimensional space; moreover, functional spaces in duality have to be dealt with correctly, but the goal here is to understand the way to define the structure-preserved pHs and see how the collocated control and observation are being adequately discretized. Note that in this part, vector calculus is being used, we make use of divergence and gradient operators.

**4) Exterior calculus representation of PH systems**

Exterior differential calculus using dierential forms is an elegant mathematical framework to describe in a coordinate-free setting many physical phenomena, e.g. (finite-dimensional) Hamiltonian dynamics of mechanical systems or (infinite-dimensional) systems of conservation or balance laws like the Maxwell equations of electrodynamics. In contrast to vector calculus, a single differential operator, the exterior derivative, is defined in which terms the operations gradrot, div can be expressed. This requires an additional operation, the Hodgestar, which relates the canonical duality pairing of differential forms and the standard (L2) inner product on an infinite-dimensional space. The sequence of spaces of differential forms, which are related by the exterior derivative, is the so-called de Rham complex. Functional spaces of dierential forms in geometric discretization shall form appropriate sub-sequences of this chain complex. We present the formulation of Stokes-Dirac structures and canonical PH systems of conservation laws on arbitrary spatial dimension in terms of exterior calculus and highlight the relations to the vector calculus setting. The concepts and operations of exterior differential calculus will be explained in this context and illustrated on the examples of the 2D wave and shallow water equations and the 3D heat equation. The exercise part focuses on (a) practical calculations to grasp the key properties of exterior calculus and the underlying physical notions and (b) on the formulation of variational problems and their transformation to vector calculus notation, which is important for the implementation of PFEM in FE software.

**5) Discrete time PHS for simulation and control**

In this last part of the course we turn to the geometric or structure-preserving time discretization of PH systems. We will introduce symplectic integration of (closed) Hamiltonian systems based on the analysis of basic examples, which serve to understand the properties like area preservation or the existence of a modied Hamiltonian. Symplectic schemes are also appropriate candidates for the discretization of (open) PH systems, and we will present a definition of discrete-time PH systems based on symplectic integration. Finally, we will show how geometric integration can also be used to define target systems for discrete-time control of sampled systems, and how such a geometric control design can improve the closed-loop behavior significantly at low sampling rates.

Ghislain Haine is Associate professor and Denis Matignon is full professor, both at the Institut Superieur de_l’Aeronautique et de l’Espace (ISAE), France.

Paul Kotyczka is a professor at the Technical University of München, Germany

This course assumes the students are familiar with the basics of Matlab, Python and Fenics.

The course will take place from Monday March 16 until Wednesday March 18, 2020. The location is Eindhoven University of Technology.

Registration fee for taking or auditing a full course is € 250. This fee is waived for DISC members. The registration form is available on the DISC course platform, or send an email to secr@disc.tudelft.nl.

Please register before March 9, 2020.

You can obtain 1 ECTS for attending the DISC Winter Course. Please note that you have to be present at all sessions to obtain the credits.