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Differential-Algebraic Equations: A Projector Based Analysis

verfasst von: René Lamour, Roswitha März, Caren Tischendorf

Verlag: Springer Berlin Heidelberg

Buchreihe : Differential-Algebraic Equations Forum

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Über dieses Buch

Differential algebraic equations (DAEs), including so-called descriptor systems, began to attract significant research interest in applied and numerical mathematics in the early 1980s, no more than about three decades ago. In this relatively short time, DAEs have become a widely acknowledged tool to model processes subjected to constraints, in order to simulate and to control processes in various application fields such as network simulation, chemical kinematics, mechanical engineering, system biology.

DAEs and their more abstract versions in infinite-dimensional spaces comprise a great potential for future mathematical modeling of complex coupled processes.

The purpose of the book is to expose the impressive complexity of general DAEs from an analytical point of view, to describe the state of the art as well as open problems and so to motivate further research to this versatile, extra-ordinary topic from a broader mathematical perspective.

The book elaborates a new general structural analysis capturing linear and nonlinear DAEs in a hierarchical way. The DAE structure is exposed by means of special projector functions. Numerical integration issues and computational aspects are treated also in this context.

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Inhaltsverzeichnis

Frontmatter

Projector based approach

Frontmatter

Chapter 1. Linear constant coefficient DAEs

Abstract

This chapter represents an introduction into the projector based framework by means of well-understood constant coefficient DAEs. In particular, we demonstrate that all components of the Kronecker structure of a regular matrix pencil can be described by so-called admissible matrix sequences and their associated projectors. We provide a complete decoupling of the DAE into its slow and fast subsystems by this technique. Thereby we do not transform the DAE itself, instead we express all system coefficients and characteristics in terms of the matrix sequence which is directly computed from the original matrix pencil. The spectral projector of the matrix pencil, for instance, results as a product of completely decoupling projectors.

René Lamour, Roswitha März, Caren Tischendorf

Chapter 2. Linear DAEs with variable coefficients

Abstract

We characterize the class of regular linear DAEs with variable coefficients by means of admissible matrix function sequences and associated projector functions. We do not use derivative arrays; instead we generate the admissible matrix functions directly from the coefficients of the given DAE. The so-called tractability index as well as several characteristic values are attributed to each regular DAE, which generalizes the Kronecker index and the Kronecker structural data of constant matrix pencils. We provide a constructive decoupling of regular DAEs into an inherent regular ODE and a subsystem containing all differentiations. Then, with this background, a comprehensive linear DAE theory is provided, including qualitative flow properties and a rigorous description of admissible excitations. Moreover, relations to several canonical forms and other index notions are addressed.

René Lamour, Roswitha März, Caren Tischendorf

Chapter 3. Nonlinear DAEs

Abstract

We discuss general nonlinear DAEs with properly involved derivative and characterize the class of regular nonlinear DAEs by means of admissible matrix function sequences and associated projector functions which are pointwise generated from the first partial Jacobians of the given DAE data, and which directly generalize those given for linear DAEs. We do not use higher derivatives and derivative arrays. We show that, in particular, all Hessenberg form DAEs and large classes of DAEs resulting from the modified nodal analysis in circuit simulation are regular in this sense. We provide new local solvability assertions and perturbation results. We further introduce so-called regularity regions of a DAE and prove a practically useful theorem concerning linearizations. Several characteristic values including the tractability index are attributed to each regularity region, which generalizes the Kronecker index and the Kronecker structural data of constant matrix pencils. It is pointed out that a DAE may have several regularity regions, also those with different characteristics. Solutions may cross the borders of these regions featuring a critical behavior.

René Lamour, Roswitha März, Caren Tischendorf

Index-1 DAEs: Analysis and numerical treatment

Frontmatter

Chapter 4. Analysis

Abstract

We investigate the structure of general nonlinear index-1 DAEs with properly involved linear derivative term. We show that each solution is actually a somewhat wrapped solution of the inherent explicit ODE. A certain implicitly given decoupling function plays its role when describing both the inherent ODE and the wrapping. With this background, we describe the set of consistent initial values, provide consistent initializations and prove local solvability and perturbation results. In turn, these results prove of value in Chapter 5 to analyze numerical integration methods and in Chapter 6 to investigate the asymptotical flow behavior.

René Lamour, Roswitha März, Caren Tischendorf

Chapter 5. Numerical integration

Abstract

On the basis of the structural characterization of index-1 DAEs given in Chapter 4, we analyze how integration methods behave when applied to those DAEs. We consider backward differentiation methods, Runge-Kutta methods and general linear methods. We concentrate on the question of whether a given method which is directly applied to the original DAE passes the wrapping unchanged and is handed over to the so-called inherent explicit ODE. The answer appears not to be a feature of the method, but a property of the DAE formulation. If the subspace accommodating the derivative term is actually time-invariant, then the integration method reaches the inherent explicit ODE unchanged. This makes the integration smooth to the extend to which it may be smooth for explicit ODEs. Otherwise one has to expect additional serious stepsize restrictions.

René Lamour, Roswitha März, Caren Tischendorf

Chapter 6. Stability issues

Abstract

We consider general nonlinear index-1 DAEs with properly involved linear derivative term and adress stability topics by means of the structural characterization of those DAEs given in Chapter 4, We investigate contractivity, dissipativity and stability in the sense of Lyapunov, and generalize the classical notions to make sense for DAEs. We provide the related solvability assertions concerning the infinite interval. In particular, an appropriately modified Lyapunov theorem results. We further discuss how integration methods reflect the respective flow properties. We point out that one can benefit from such DAE formulations which show a time-invariant subspace accommodating the derivative term.

René Lamour, Roswitha März, Caren Tischendorf

Computational aspects

Frontmatter

Chapter 7. Computational linear algebra aspects

Abstract

This chapter is devoted to computational aspects of the practical preparation of all ingredients of admissible matrix function sequences and the associate projectors. Eventually one has to carry out matrix factorizations, rank calculations, and generalized inverses. The characteristic values as well as the tractability index arise as byproducts of certain matrix factorizations. We provide several versions to accomplish the basic step of the matrix function sequence from one level to the next. These tools apply to general rectangular DAEs. Thereby, from the numerical viewpoint, the so-called widely orthogonal projector functions prove to be favorable. Further, a special more involved algorithm is developed for regular DAEs. The provided algorithms can be applied to check regularity and to monitor the characteristic values and the index.

René Lamour, Roswitha März, Caren Tischendorf

Chapter 8. Aspects of the numerical treatment of higher index DAEs

Abstract

This chapter sheds light on aspects of the direct numerical treatment of higher index DAEs. Not surprisingly, the integration methods approved for regular index-1 DAEs not longer perform well or fail, if they are applied in the same way to general higher index DAEs. We demonstrate various troubles, e.g., when integrating linear index-3 DAEs. These instabilities and numerical difficulties are due to the ill-posed character of the DAE solutions with respect to perturbations. We do not report index reduction techniques and methods basically exploiting special structural peculiarities to counteract the difficulties, which are discussed by many other authors. We present a procedure for the practical calculation of the index and add a few remarks on consistent initialization in the higher index case. This is of importance for users of DAE solver packages since they usually require knowledge about the DAE index.

René Lamour, Roswitha März, Caren Tischendorf

Advanced topics

Frontmatter

Chapter 9. Quasi-regular DAEs

Abstract

Regularity of DAEs as described in Part I is supported by several constant-rank conditions. In the present chapter we relax these conditions and allow for rank changes. Replacing the nullspaces of the admissible matrix functions by continuous subnullspaces we again obtain continuous matrix function sequences. These modified sequences inherit most of the properties given for the standard sequences in Part I. By this, we figure out so-called quasi-regular DAEs. In particular, linear DAEs that are transformable into so-called standard canonical form belong to quasi-regular DAEs. A decoupling procedure applied to general linear quasi-regular DAEs is offered. Unfortunately, the rank values serving as characteristic values within the regularity concept in Part I now lose their meaning. Similarly to the mostly applied differentiation index notion, quasi-regularity appears to be somewhat diffuse. We touch on difficulties arising with the use of subnullspaces.

René Lamour, Roswitha März, Caren Tischendorf

Chapter 10. Nonregular DAEs

Abstract

We deal with DAEs having a properly involved linear derivative term. The DAE is possibly rectangular comprising different numbers of equations and unknowns. A DAE not being regular is said to be nonregular. We touch on this topic just slightly and emphasize the scope of different interpretations by several case studies. The framework of sequences of admissible projector functions and associated projector functions applies in the same way as in Part I. We offer a generalization of the tractability index for the nonregular case. We further provide a projector based decoupling of linear DAEs and characterize consistency conditions and those components of the unknown which can be chosen arbitrarily.

René Lamour, Roswitha März, Caren Tischendorf

Chapter 11. Minimization with constraints described by DAEs

Abstract

The present chapter collects results obtained by means of the projector based approach to DAEs, which are relevant in view of optimization. We do not at all undertake to offer an overview concerning the large field of control and optimization with DAE constraints and do not touch the huge arsenal of direct minimization methods. We address the basic topics of adjoint and self-adjoint DAEs and discuss the Hamiltonian property. We provide a necessary extremal conditions for the case of a nonlinear cost and a nonlinear DAE constraint. Necessary and sufficient extremal conditions are given for linear-quadratic problems. Moreover, an appropriate generalization of the Riccati feedback solution is developed. In each part, we direct particular attention to the properties of the resulting optimality DAE. If one intends to apply indirect optimization, then one should take great care to ensure good properties, such as regularity with index 1, in advance by utilizing the scope of modeling. By providing criteria in terms of the original problem data we intend to assist specialists in modeling.

René Lamour, Roswitha März, Caren Tischendorf

Chapter 12. Abstract differential-algebraic equations

Abstract

The concept of regular DAEs developed in Part I for DAEs in finite-dimensional spaces is generalized to some extend for DAEs acting in Hilbert spaces, which are called abstract differential-algebraic equations (ADAEs). Such a framework aims to provide a systematic approach for coupled systems of different type. It should be emphasized that this working field is still in its infancy and further research is absolutely reasonable. After having discussed various special cases we turn to a class of linear ADAEs which covers parabolic PDEs and index-1 DAEs as well as couplings thereof. We treat this class in detail by means of Galerkin methods yielding an existence and uniqueness result for the ADAE as well as an error estimation for the perturbed systems.

René Lamour, Roswitha März, Caren Tischendorf

Backmatter

Titel: Differential-Algebraic Equations: A Projector Based Analysis
verfasst von: René Lamour
Roswitha März
Caren Tischendorf
Copyright-Jahr: 2013
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-27555-5
Print ISBN: 978-3-642-27554-8
DOI: https://doi.org/10.1007/978-3-642-27555-5

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