## Seminar in Numerical Analysis: Alfio Borzi (Universität Würzburg)

**The Liouville equation, its extensions, and related optimal control problems**

The Liouville equation is the fundamental building block of models that govern the evolution of density functions of multi-particle systems. These models include different Fokker-Planck and Boltzmann equations that arise in many application fields ranging from gas dynamics to pedestrians' motion where the need arises to control these systems.

This talk provides an introduction to the formulation and solution of optimal control problems governed by the Liouville equation and related models. The purpose of this framework is the design of robust controls to steer the motion of particles, pedestrians, etc., where these agents are represented in terms of density functions. For this purpose, expected-value cost functionals are considered that include attracting potentials and different costs of the controls, whereas the control mechanism in the governing models is part of the drift or is included in a collision term.

In this talk, theoretical and numerical results concerning ensemble optimal control problems with Liouville, Fokker-Planck and linear Boltzmann equations are presented.

For further information about the seminar, please visit this webpage.

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