# Minisymposium “Optimal Control of Partial Differential Equations”

#### Organizer: Prof. Dr. Martin Gugat

(Tuesday, **03.03**.20, 10:20-12:20, lecture hall H12)

**Prof. Dr. Alfio Borzi (University of Würzburg):**

“*On ensemble optimal control problems governed by Liouville, Fokker-Planck and linear Boltzmann equations*”

In this talk, a survey of recent results on the optimal control of ensembles governed by the Liouville (continuity), Fokker-Planck and linear Boltzmann equations is presented. The purpose of this frame-work is the design of robust controls of ensembles of dynamical systems that model the motion of massive particles, pedestrians, etc., where these agents are represented in terms of density functions. In this statistical framework, 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 the collision term. Some theoretical and numerical results concerning the cases of deterministic and stochastic motion of non-interacting agents and of massive particles subject to collision with smaller particles are illustrated.

This is joint work with M. Annunziato (Salerno), J. Bartsch (Würzburg), F. Fanelli (Lyon), G. Nastasi and V. Romano (Catania), S. Roy (Texasat Arlington).

**Prof. Dr. Michael Hinze (University of Koblenz):**

“*Fluid mechanic shape optimization with phase field models*”

We consider a phase field approach to shape and topology optimization in fluid flow. The mathematical modeling leads to a PDE constrained optimization problem with control in the coefficients where the control enters as phase field in the Darcy term of the Navier-Stokes model. We prove existence of solutions and present a numerical realization based on the finite element method. We illustrate the performance of our approach with some numerical examples.

This is joint work with Harald Garcke (University Regensburg) and Christian Kahle (University of Koblenz-Landau).

**Prof. Dr. Boris Vexler (Technical University of München):**

“*Numerical Analysis of sparse initial data identification for parabolic problems*”

We consider a problem of identification of initial data for a homogeneous parabolic equation from an observation of the final state.

In general, such problems are known to be exponentially ill-posed. We are interested in the situation, where the initial data, we are looking for, is known to be sparse, i.e. to have a support of Lebesgue measure zero. The strong smoothing property of parabolic equations makes it difficult to identify such sparse initial data. The remedy is the incorporation of the information that the unknown initial data should be sparse into the optimal control formulation. This allows for robust identification of initial data consisting of finitely many Dirac impulses with unknown number and positions. For the resulting optimal control problem we discuss finite element discretization and provide precise error analysis.

**Prof. Dr. Simone Göttlich (University of Mannheim)**:

“*Energy supply systems: state-of-the-art and challenges*”

The coupling of different transport dynamics is a challenging field and becomes even more important in the near future when air taxis will supplement the city traffic or the energy transition requires new storage systems. Therefore, a mathematical framework for the coupling of gas networks to electric grids is presented to describe in particular the transition from gas to power. The dynamics of the gas flow are given by the isentropic Euler equations, while the power flow equations are used to model the power grid. For simulation purposes, we apply appropriate numerical methods and show in an experimental study how gas-to-power might influence the dynamics of the gas and power network, respectively. We also present a framework with a compressor station to control the gas pressure such that certain bounds are satisfied. The numerical results show how fast in the demand lead to necessary operator actions.

This is joint work with E. Fokken and O. Kolb (both University of Mannheim).

**Prof. Dr. Falk Hante (Humboldt University of Berlin):**

“*Decision making in optimal control of complex dynamical systems*”

In the age of globalization and digitalization the optimization of systems with real and virtually linked components, with elements of discrete logic and adaptive models with hierarchies up to partial differential equations increasingly gain importance. These challenges occur for instance in laying out sustainable energy networks, in coordinating autonomous traffic or in the control of cell dynamics in human neural networks. In this talk I will present a decomposition approach combining mixed-integer programming techniques and methods from optimal control in order to solve such mixed-integer optimal control problems efficiently.