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Collaboration Announcement

Feedback Control of Kinetic and Hyperbolic Equations

Jun 20 - Jul 31, 2015

Department of Mathematics, Aachen
Department of Mathematics

Templergraben 59 D-52056
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For applications in gas dynamics, shallow-water equations as well as re-entry problems in computational fluid dynamics stabilization of flow patterns using for example boundary data is of importance. Recently, there have been efforts in the development of analytical tools to design exponentially fast stabilizing boundary conditions using Lyapunov functions. Clearly, numerical schemes should recover similar rates for stabilization problems. The question of suitable high-order finite volume discretizations realizing the desired stability property will be investigated within this proposal. In particular, we derive suitable discretizations of the boundary data such that corresponding discrete Lyapunov functions decay exponentially fast. We want to prove exponential stability as well as high order resolution of the stabilized state. Starting point of the investigation will be the 1-d isentropic Euler equations including friction discretized by finite-volume methods. Up to the discretization of the source terms those result should then also extend to shallow-water equations. The extension of the control concepts to the Lattice Boltzmann equation is of importance in order to treat the topic theoretically and numerically. Here, the main difficulty arises in suitable estimates of the stiff source term that may be in conflict with the exponential convergence.

Analytical results for the stabilization of initial-boundary value hyperbolic problems have been established mostly in the realm of classical solutions, called semi-global solutions due to a finite time of existence, and only for a limited class of source terms. The influence of the source term is of immanent importance in these applications since it, for example, determines the equilibria. Analytical investigations and the construction feedback laws for stabilization rely in most cases on the construction of suitable strict Lyapunov functions. Exponential decay of small perturbations to equilib


The focus of this research will be on the derivation and numerically discretization of feedback boundary conditions for stabilization of hyperbolic balance laws inheriting the continuous properties and to establish exponential decay of perturbations.  The approach is based on Lattice Boltzmann methods and relaxation schemes.


Martin FrankRWTH Aachen University
Axel HaeckRWTH Aachen University, Germany
Michael HertyRheinisch Westfälische Technische Hochschule Aachen
Wen-An YongTsinghua University


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Department of Mathematics, Aachen (Aachen)
Templergraben 59 D-52056