Roswitha März

Differential-algebraic equations : a projector based analysis

Notations.- Introduction.- Part I. Projector based approach.- 1 Linear constant coefficient DAEs.-.2 Linear DAEs with variable coefficients.- 3 Nonlinear DAEs.- Part II. Index-1 DAEs: Analysis and numerical treatment.- 4 Analysis.- 5 Numerical integration.- 6 Stability issues.- Part III. Computational aspects.- 7 Computational linear algebra aspects.- 8 Aspects of the numerical treatment of higher index DAEs.- Part IV. Advanced topics.- 9 Quasi-regular DAEs.- 10 Nonregular DAEs.- 11 Minimization with constraints described by DAEs.- 12 Abstract differential algebraic equations.- A. Linear Algebra - Basics.-.B. Technical Computations.- C Analysis.- References.- Index.

Numerical methods for differential algebraic equations

Differential algebraic equations (DAE) are special implicit ordinary differential equations (ODE) where the partial Jacobian f ′ y ( y, x, t ) is singular for all values of its arguments.

The index of linear differential algebraic equations with properly stated leading terms

For linear differential-algebraic equations with properly stated leading terms and coefficients that are just continuous an index notion is introduced. The index criteria are given in terms of the original coefficients. The index is shown to be invariant under regular transformations of the unknown function, but also under refactorizations of the leading term. Inherent regular explicit ordinary differential equations are described in detail.

Canonical projectors for linear differential algebraic equations

Abstract This paper aims at improving the decoupling projector chain approach for DAEs to provide complete decouplings. These special canonical projectors are shown to form exactly the spectral projection related to the matrix coefficient pair given by the DAE. Further, they prove their value when describing the constraint manifolds of the DAE.

A Unified Approach to Linear Differential Algebraic Equations and their Adjoints

Instead of a single matrix occurring in the standard setting, the leading term of the linear differential algebraic equation is composed of a pair of well matched matrices. An index notion is proposed for the equations. The coefficients are assumed to be continuous and only certain subspaces have to be continuously differentiable. The solvability of lower index problems is proved. The solution representations are based on the solutions of certain inherent regular ordinary differential equations that are uniquely determined by the problem data. The assumptions allow for a unified treatment of the original equation and its adjoint. Both equations have the same index and are solvable simultaneously. Their fundamental solution matrices satisfy a relation that generalizes the classical Lagrange identity.

Differential algebraic systems anew

It is proposed to figure out the leading term in differential algebraic systems more precisely. Low index linear systems with those properly stated leading terms are considered in detail. In particular, it is asked whether a numerical integration method applied to the original system reaches the inherent regular ODE without conservation, i.e., whether the discretization and the decoupling commute in some sense. In general one cannot expect this commutativity so that additional difficulties like strong stepsize restrictions may arise. Moreover, abstract differential algebraic equations in infinite-dimensional Hilbert spaces are introduced, and the index notion is generalized to those equations. In particular, partial differential algebraic equations are considered in this abstract formulation.

Stability preserving integration of index-1 DAEs

For index-1 DAEs with properly stated leading term, we characterize dissipative and contractive flows and study how the qualitative properties of the DAE solutions are reflected by numerical approximations. The best situation occurs when the discretization and the decoupling procedure commute. It turns out that this is the case if the relevant part of the inherent regular ODE has a constant state space. Different kinds of reformulations are studied to obtain this property. Those reformulations might be expensive, hence, in order to avoid them, criteria ensuring the given DAE to be numerically equivalent to a numerically qualified representation are proved.

On Linear-Quadratic Optimal Control Problems for Time-Varying Descriptor Systems

We deal with linear-quadratic optimal control problems for time-varying descriptor systems in a Hilbert space setting. A sufficient solvability condition is given by means of an appropriately stated linear boundary value problem (BVP) and by discussing the special structure of the regular differential system inherent in this BVP.

Differential Algebraic Systems with Properly Stated Leading Term and MNA Equations

Differential algebraic equations with properly stated leading term are equations of the form A(x(t),t)(d(x(t),t))’ + b(x(t),t) = 0 with in some sense well-matched coefficients. Systems resulting from the modified nodal analysis (MNA) in circuit simulation promptly fit into this form.

On linear differential-algebraic equations and linearizations

On the background of a careful analysis of linear DAEs, linearizations of nonlinear index-2 systems are considered. Finding appropriate function spaces and their topologies allows to apply the standard Implicit Function Theorem again. Both, solvability statements as well as the local convergence of the Newton-Kantorovich method (quasilinearization) result immediately. In particular, this applies also to fully implicit index-1 systems whose leading nullspace is allowed to vary with all its arguments.