Numerik-Oberseminar SS 2001

Terminplanung der Vorträge:

Abstracts zu den Vorträgen:

Abstract:
Energy dissipation is introduced via linear velocity feedbacks acting through a portion of the boundary as traction forces, resulting in exponential decay. This is achieved without the imposition of strong geometric assumptions on the controlled portion of the boundary through the use of a physically viable feedback which is only a function of velocity, as opposed to also containing the tangential derivative of the displacement, as has been seen in earlier work. Proof is based on the "multiplier method" and relies critically on sharp trace estimates for the tangential derivative of the displacement on the boundary as well as on unique continuation results for the corresponding static system.

Abstract:
Viele prominente Probleme aus der Kontrolltheorie und aus der linearen Algebra sind von algebraisch geometrischer Natur. Als Beispiel moechten wir das Problem der Polvorgabe so wie das Matrizen-Komplettierungsproblem erwaehnen. Auf der geometrischen Seite entsprechen diesen Problemen oft Fragestellungen aus dem Schubertkalkuehl. In unserem Vortrag geben wir eine Uebersicht ueber Resultate die mittels algebraisch geometrischen Methoden erreicht wurden.

Abstract:
Gekoppelte Systeme von Differentialgleichungen treten in vielen Anwendungen auf. Im Vortrag werden zwei Problemklassen herausgegriffen, zum einen flexible Mehrkörpersysteme in der Fahrzeug- und Maschinendynamik und zum anderen inelastische Verformungen von Kontinua in der Materialforschung. Beide Male liegt ein gekoppeltes System von partiellen Differentialgleichungen und differential-algebraischen Gleichungen vor. Zur numerischen Lösung werden unter anderem Techniken aus der Ortsdiskretisierung von Sattelpunktproblemen und aus der Zeitintegration mit Runge-Kutta-Verfahren eingesetzt. Verschiedene Simulationsbeispiele erläutern die numerischen Methoden.

Abstract:
In this talk a new index reduction technique is discussed for the treatment of large scale differential-algebraic systems for which extra structural information is available. Based on this information reduced derivative arrays are formed and instead of using expensive subspace computations the index reduction is obtained by introducing new variables. The new approach is demonstrated for several important classes of differential-algebraic systems, where the structural information is available. These include multibody systems, circuit simulation problems and semidiscretized Navier-Stokes equations. The effectiveness of the new approach is demonstrated via numerical examples.

Abstract:
The talk is divided in two parts. First, the general affine invariance concept for Newton's method is presented, and its implications on convergence theory and algorithmic robustness is discussed. Subsequently, an affine invariant norms is constructed for the special case of equality constrained optimization problems.

In the second part, a function space complementarity method is applied to optimal control problems. The advocated approach combines ideas of both direct and indirect methods and realizes a nested reduction of mesh size and duality gap. An affine invariant inexact continuation method is used for following the central path towards the solution. The successful solution of a well-known difficult optimal control problem documents the effectivity of the approach.

Abstract:
Although multistep methods with variable step sizes are widely used in numerical computations, their analysis is still not complete. Because of the non-uniform grid, non-constant coefficients appear in the resulting scheme. Theoretical tools developed for difference equations with constant coefficients are therefore not applicable.
Among the abundance of methods, the backward differentiation formulae (BDF) seem to be of particular interest. In this talk, different stability properties of the two-step BDF are discussed. It turns out that the step size ratios need to be suitably bounded in order to prove stability.
New results can be presented for the time discretisation of abstract semilinear parabolic equations with a moderate nonlinearity: Stability as well as optimal smooth-data error estimates are derived for step size ratios bounded by 1.91.
The talk will be supplemented by R. D. Grigorieff with remarks on the stability of the discretisation of semigroups by the two-step BDF and on stability and error estimates for the three-step BDF applied to a parabolic problem.

Abstract:
The class of normal matrices, which arises naturally in the context of unitary equivalence, is important throughout matrix analysis and generalizes unitary, Hermitian, and skew-Hermitian matrices. Analogous to the positive definite case, one can also define normal matrices in indefinite inner product spaces to be matrices that commute with their adjoints. It turns out that this definition leads to some difficulties: in contrast to the positive definite case, it is impossible to obtain a complete classification of normal matrices. From this point of view, the class of normal matrices in indefinite inner product spaces is `too big'.
However, there is a long list of conditions that are equivalent to normality in the positive definite case. In the indefinite case, they are usually non-equivalent, i.e. the conditions may be stronger or weaker than the condition of normality. Thus, these conditions define classes of matrices that are `more or less' normal. In this talk, these classes are examined with the aim of obtaining a class of normal matrices that still generalizes unitary, Hermitian, and skew-Hermitian matrices and that is not `too big'.

Edmond A. Jonckheere, University of Southern California, Los Angeles, CA, USA
Linear Dynamically Varying Control of Nonlinear Systems over Compact Sets
Do 19.7.2001, 16:15 Uhr in MA 144

Abstract:
Nonlinear systems running over compact sets are well known to have attractors containing a rich set of trajectories. This allows a variety of control objectives to be achieved by selecting a trajectory and forcing the controlled system to track the reference trajectory via a small perturbational control, despite offset in initial conditions, model uncertainty, extraneous disturbances, etc. The Linear Dynamically Varying (LDV) control scheme consists in linearizing the tracking error about the nominal trajectory and forcing it to go to zero via either LQ or H-Infinity techniques. The central mathematical object of concern of this approach is an algebraic Riccati equation of the functional type. Surprisingly, the solution $X_\theta$, which provides the cost to stabilize an orbit starting at $\theta$, is continuous, which means that the cost of tracking a periodic orbit is almost the same as the cost of tracking an aperiodic orbit starting at a nearby point. A numerical scheme for solving the functional Riccati equation relying on the ergodic property of recurrence will be developed. It will be shown that the numerical scheme is numerically stable and well conditioned.

Ref: S. Bohacek and E. A. Jonckheere, ``Linear Dynamically Varying LQ control of nonlinear systems over compact sets,'' IEEE Transaction on Automatic Control, vol. 46, pp. 840-852, June 2001.

Leiba Rodman, College of William and Mary, Williamsburg, VA, USA
Almost periodic factorizations of matrix functions
Di 21.8.2001, 16:15 Uhr in MA 750

Abstract:
An almost periodic (AP) factorization of an almost periodic (in the classical sense of Bohr) function f(t) has the form $f(t)=f_+(t)e^{i\lambda t}f_-(t)$, where $f_+(t)$ (resp. $f_-(t)$) is an almost periodic function that is bounded away from zero and, together with its inverse, admits an analytic continuation into the open upper (resp. lower) halfplane. The AP factorization is known for a long time, and its existence, under the natural hypothesis that f(t) itself is bounded away from zero, is a key property of almost periodic functions.

Recently, there is a renewed interest in the AP factorization, for more general classes of matrix valued almost periodic functions of one and of several real variables. The interest is motivated by several applications, some of which will be mentioned.

The talk will focus on a review of some recent works and results in this area, starting with the simpler and more thoroughly studied case of periodic factorization. Open problems will be stated.