Peter Kunkel

Canonical forms for linear differential-algebraic equations with variable coefficients

Abstract We give a new set of local characterizing quantities for the treatment of linear differential-algebraic equations with variable coefficients. This leads to new global invariances under which we can find a generalization of the global (or differentiation) index and get a new existence and uniqueness theorem.

Efficient numerical path following beyond critical points

The paper presents a numerical pathfollowing technique based on an implicit change of parametrization using QR-decomposition. An automatic steplength control for tangent continuation is derived on a theoretical basis. In the suggested approach, turning points do not play any exceptional role. For the actual computation of (simple) bifurcation points, a special algorithm based on the augmented system of Moore is developed. This treatment includes imperfect bifurcations and permits an easy way of branch switching. An efficient linear equation solver is worked out in connection with the QR-decomposition. Finally, numerical comparisons are included that clearly document the efficiency of the newly developed algorithms.

Analysis of Over- and Underdetermined Nonlinear Differential-Algebraic Systems with Application to Nonlinear Control Problems

Abstract. We study over- and underdetermined systems of nonlinear differential-algebraic equations. Such equations arise in many applications in circuit and multibody system simulation, in particular when automatic model generation is used, or in the analysis and solution of control problems in the behavior framework.¶We give a general (local) existence and uniqueness theory and apply the results to analyze when nonlinear implicit control problems can be made regular by state or output feedback.¶The theoretical analysis also leads immediately to numerical methods for the simulation as well as the construction of regularizing feedbacks.

Analysis and Numerical Solution of Control Problems in Descriptor Form

Abstract. We study linear variable coefficient control problems in descriptor form. Based on a behaviour approach and the general theory for linear differential algebraic systems we give the theoretical analysis and describe numerically stable methods to determine the structural properties of the system like solvability, regularity, model consistency and redundancy. We also discuss regularization via feedback.

A New Software Package for Linear Differential-Algebraic Equations

We describe the new software package GELDA for the numerical solution of linear differential-algebraic equations with variable coefficients. The implementation is based on the new discretization scheme introduced in [P. Kunkel and V. Mehrmann, SIAM J. Numer. Anal., 33 (1996), pp. 1941--1961]. It can deal with systems of arbitrary index and with systems that do not have unique solutions or inconsistencies in the initial values or the inhomogeneity. The package includes a computation of all the local invariants of the system, a regularization procedure, and an index reduction scheme, and it can be combined with every solution method for standard index-1 systems. Nonuniqueness and inconsistencies are treated in a least square sense. We give a brief survey of the theoretical analysis of linear differential-algebraic equations with variable coefficients and discuss the algorithms used in GELDA. Furthermore, we include a series of numerical examples as well as comparisons with results from other codes, as far as this is possible.

The linear quadratic optimal control problem for linear descriptor systems with variable coefficients

We study linear quadratic optimal control problems for linear variable coefficient descriptor systems. Generalization from the case of standard control problems leads to several difficulties. We discuss a behavioral approach that solves some of these difficulties. Furthermore, an analysis of general rectangular systems is given and generalized Euler-Lagrange equations and Riccati differential algebraic equations are discussed.

Regularization of Linear Descriptor Systems with Variable Coefficients

We study linear descriptor control systems with rectangular variable coefficient matrices. We introduce condensed forms for such systems under equivalence transformations and use these forms to detect whether the system can be transformed to a uniquely solvable closed loop system via state or derivative feedback. We show that under some mild assumptions every such system consists of an underlying square subsystem that behaves essentially like a standard state space system, plus some solution components that are constrained to be zero.

Optimal control for unstructured nonlinear differential-algebraic equations of arbitrary index

We study optimal control problems for general unstructured nonlinear differential-algebraic equations of arbitrary index. In particular, we derive necessary conditions in the case of linear-quadratic control problems and extend them to the general nonlinear case. We also present a Pontryagin maximum principle for general unstructured nonlinear DAEs in the case of restricted controls. Moreover, we discuss the numerical solution of the resulting two-point boundary value problems and present a numerical example.

A New Class of Discretization Methods for the Solution of Linear Differential-Algebraic Equations with Variable Coefficients

We discuss new discretization methods for linear differential-algebraic equations (DAEs) with variable coefficients. We introduce numerical methods to compute the local invariants of such DAEs that were introduced by the authors in a previous paper [P. Kunkel and V. Mehrmann, J. Comput. Appl. Math., 56 (1994), pp. 225--251]. Using these quantities we are able to determine numerically global invariances like the strangeness index, which generalizes the differentiation index for DAEs that in particular include undetermined solution components. Based on these methods we then obtain regularization schemes, which allow us to employ general solution methods. The new methods are tested on a number of numerical examples.

Smooth factorizations of matrix valued functions and their derivatives

SummaryIn this paper we study the numerical factorization of matrix valued functions in order to apply them in the numerical solution of differential algebraic equations with time varying coefficients. The main difficulty is to obtain smoothness of the factors and a numerically accessible form of their derivatives. We show how this can be achieved without numerical differentiation if the derivative of the given matrix valued function is known. These results are then applied in the numerical solution of differential algebraic Riccati equations. For this a numerical algorithm is given and its properties are demonstrated by a numerical example.