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# Mini courses

The following two mini courses will be held during MTNS 2022. Participation is free of charge for all MTNS 2022 attendees without additional fee.

## Mini Course on Measure differential equations: modeling and numerical solution

### organised by Benoît Bonnet, Didier Henrion, Swann Marx, and Francesco Rossi

This mini-course focuses on numerical methods for solving differential equations on measures, with applications in optimization and control, as well as modeling multi-scale multi-agent systems. Covered material include the moment sums of squares hierarchy and Eulerian schemes.

Part one - modeling: In the first part of the mini-course we describe the optimization and control problems that can be modelled in terms of partial differential equations (PDEs) on probability measures. We present the main existing wellposedness theories which are based on optimal mass transportation, establish the correspondence between non-linear ordinary differential equations (ODEs) and linear transport PDEs, and discuss some of the recent progresses made in this area of the literature. We also present a large number of domains in which measure differential dynamics have been successfully applied, e.g. multi-anticipative road traffic, crowd dynamics, collective behavior in animal groups, supply chains, age-structured populations.

Part two - solving: In the second part of the mini-course we introduce some numerical methods for solving measure PDEs. On the one hand, we describe briefly the moment sums of squares a.k.a. Lasserre hierarchy, originally introduced for polynomial optimization, and later on extended to optimal control of ODEs and more recently to numerical computation of non-linear PDEs. On the other hand, we describe Euler schemes for measure differential equations with non-local terms. The idea is to discretize the measure into a grid, and to let it evolve following some adapted ODE. Convergence is based on optimal transportation strategies that provide a metric tool (the Wasserstein distance) for the study of measure dynamics.

Confirmed speakers:

• Nastassia Pouradier-Duteil - continuum limits of collective dynamics with time-varying weights
• Benoît Bonnet - differential inclusions in measure spaces
• Claudia Totzeck - optimization methods for multi-agent systems
• Didier Henrion - tutorial on the moment-SOS aka Lasserre hierarchy
• Swann Marx - conservation laws solved with the Lasserre hierarchy and the Christoffel-Darboux polynomial

## Mini Course on Infinite-dimensional port-Hamiltonian systems

### organised by Birgit Jacob and Timo Reis

The theory of port-Hamiltonian systems provides a geometric modelling framework for systems of various physical domains, such as mechanics, electrodynamics and thermodynamics. This approach has its roots in analytical mechanics and starts from the principle of least action, and proceeds, via the Euler-Lagrange equations and the Legendre transform, towards the Hamiltonian equations of motion. This class is further closed under network interconnection. That is, coupling of port-Hamiltonian systems again leads to a port-Hamiltonian system, whence it further allows to describe multi-physical systems, i.e., systems obtained by interaction of several physical domains. The port-Hamiltonian approach further allows the qualitative solution behavior, since it provides an energy balance. Modelling of port-Hamiltonian dynamics may result in various different types of equations, such as ordinary differential equations, differential-algebraic equations, partial differential-algebraic equations and partial differential equations. The latter two types can be reformulated as infinite-dimensional systems, which results in a need for a wide theory of infinite-dimensional port-Hamiltonian systems.

The aim of this mini-course is to give a tutorial on the theory and practice of infinite-dimensional port-Hamiltonian systems. We will provide basics of modelling, analysis and numerics for this class. In particular, we will treat the following questions in the course:

• What are practical examples of infinite-dimensional port-Hamiltonian systems?
• How is modelling of physical systems by infinite-dimensional port-Hamiltonian systems been done?
• What is known about analysis of infinite-dimensional port-Hamiltonian systems?
• What are appropriate numerical tools for infinite-dimensional port-Hamiltonian systems?
• What are open problems for infinite-dimensional port-Hamiltonian systems?

Confirmed speakers:

• Timo Reis – From finite to infinite-dimensional systems
• Bernhard Maschke – Boundary port-Hamiltonian systems
• Hans Zwart – Analysis for boundary port-Hamiltonian systems
• Paul Kotyczka – From infinite to finite-dimensional systems
• Birgit Jacob – Further results and open problems

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