## Berichte der Arbeitsgruppe Technomathematik (AGTM Report)

86

We consider two transmission boundary-value problems for the time-harmonic Maxwell equations without displacement currents. For the first problem we use the continuity of the tangential parts of the electric and magnetic fields across material discontinuities as transmission conditions. In the second case the continuity of the tangential components of the electric field E is replaced by the continuity of the normal component of the magnetization B=müH. For this problem existence of solutions is already shown in [6]. If the domains under consideration are not simply connected the solution is not unique. In this paper, we improve the regularity results obtained in [6] and then prove existence and uniqueness theorems for the first problem by extracting its solution out of the set of all solutions of the second problem. Thus we establish a connection between the solutions corresponding to the different transmission boundary conditions.

85

We consider a transmission boundary-value problem for the time-harmonic Maxwell equations without displacement currents. As transmission conditions we use the continuity of the tangential parts of the magnetic field H and the continuity of the normal components of the magnetization B=müH. This problem, which is posed over all IR3, is then restricted to a bounded domain by introducing artificial boundary conditions. We present uniqueness and existence proofs for this problem using an integral equation approach and compare the results with those obtained in the unbounded case.

84

We consider a transmission boundary-value problem for the time-harmonic Maxwell equations neglecting displacement currents. The usual transmission conditions, which require the continuity of the tangential components of the electric and magnetic fields across boundaries are slightly modified. For this new problem we show that the uniqueness of the solution depends on the topological properties of the domains under consideration. Finally we obtain existence results by using a boundary integral equation approach.

83

The paper presents theoretical and numerical investigations on simulation methods for the Boltzmann equation with axisymmetric geometry. The main task is to reduce the computational effort by taking advantage of the symmetry in the solution of the Boltzmann equation.; The reduction automatically leads to the concept of weighting functions for the radial space coordinate and therefore to a modified Boltzmann equation. Consequently the classical simulation methods have to be modified according to the new equation.; The numerical results shown in this paper - rarefied gas flows around a body with axisymmetric geometry - were done in the framework of the European space project HERMES.

82

Diffeomorphisms are given between different subsets of linear systems of fixed McMillan degree. The sets considered are the set of all systems of fixed McMillan degree, the subset of stable systems, the subset of bounded real systems, the subset of positive real systems, the subset of stable systems with Hankel singular values bounded by one. State space techniques are used in the proofs.

81

The polynomial approach introduced in Fuhrmann [1991] is extended to cover the crucial area of AAK theory, namely the characterization of zero location of the Schmidt vectors of the Hankel operators. This is done using the duality theory developed in that paper but with a twist. First we get the standard, lower bound, estimates on the number of unstable zeroes of the minimal degree Schmidt vectors of the Hankel operator. In the case of the Schmidt vector corresponding to the smallest singular the lower bound is in fact achieved. This leads to a solution of a Bezout equation. We use this Bezout equation to introduce another Hankel operator which have singular values that are the inverse of the singular values of the original Hankel operator.

80

79

Given a proper antistable rational transfer function g, a balanced realization of g is contructed as a matrix representation of the abstract shift realization introduced in Fuhrmann [1976]. The required basis is constructed as a union of sets of polynomials orthogonal with respect to weights given by the square of the absolute values of minimal degree Schmidt vectors of the corresponding Hankel operators. This extends results of Fuhrmann [1991], obtained in the generic case.

78

Several topological necessary conditions of smooth stabilization in the large have been obtained. In particular, if a smooth single-input nonlinear system is smoothly stabilizable in the large at some point of a connected component of equilibria set, then the connected component is to be an unknoted, unbounded curve.

77

A multiparameter, polynomial feedback strategy is introduced to solve the universal adapative tracking problem for a class of multivariable minimum phase system and reference signals generated by a known linear time-invariant differential equation. For 2-input, 2-output, minimum phase systems (A,B,C) with det(CB)0, a different polynomial tracking controller is given which does not invoke a spectrum unmixing set.