René Lamour

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.

The computation of consistent initial values for nonlinear index-2 differential–algebraic equations

The computation of consistent initial values for differential–algebraic equations (DAEs) is essential for starting a numerical integration. Based on the tractability index concept a method is proposed to filter those equations of a system of index-2 DAEs, whose differentiation leads to an index reduction. The considered equation class covers Hessenberg-systems and the equations arising from the simulation of electrical networks by means of Modified Nodal Analysis (MNA). The index reduction provides a method for the computation of the consistent initial values. The realized algorithm is described and illustrated by examples.

A well-posed shooting method for transferable DAE's

SummaryConsider a TPBVP for transferable nonlinear DAEs. In general the shooting equation has a singular Jacobian. A multiple shooting method which has a nonsingular Jacobian and also produces consistent initial values for the integration is presented. The estimation of the condition of the Jacobian shows the well-posedness of the method. Some illustrative examples are given

Expressing optimal control problems as differential algebraic equations

The purpose of this paper is to present an approach to express certain types of optimal control problems in terms of a system of differential algebraic equations (DAEs). This system is obtained using calculus of variations to get the Kuhn–Tucker conditions. The inequalities associated with the complementarity conditions are converted to equalities by the addition of a new variable. Such systems of DAEs are well known in the Chemical Engineering literature, and there are a number of established numerical methods for their solution. Also, we introduce here the concept of the tractability index as a general purpose way of determining the index, by establishing which part of the system of DAEs must be differentiated and how many times. This provides a systematic way of determining the index, without needing to differentiate the whole system. Numerical examples from Chemical Engineering are used to illustrate the methodology.

How Floquet-theory applies to differential-algebraic equations

Local stability of periodic solutions is established by means of a correspond ing Floquet theory for index di erential algebraic equations For this linear di erential algebraic equations with periodic coe cients are considered in detail and a natural notion of the monodromy matrix is gured out which generalizes the well known case of regular ordinary di erential equations

Multiple shooting using a dichotomically stable integrator for solving differential-algebraic equations

In previous work by the first author, it has been established that a dichotomically stable discretization is needed when solving a stiff boundary-value problem in ordinary differential equations (ODEs), when sharp boundary layers may occur at each end of the interval. A dichotomically stable implicit Runge-Kutta method, using the 3-stage, fourth-order, Lobatto IIIA formulae, has been implemented in a variable step-size initial-value integrator, which could be used in a multiple-shooting approach.In the case of index-one differential-algebraic equations (DAEs) the use of the Lobatto IIIA formulae has an advantage, over a comparable Gaussian method, that the order is the same for both differential and algebraic variables, and there is no need to treat them separately.The ODE integrator (SYMIRK [R. England, R.M.M. Mattheij, in: Lecture Notes in Math., Vol. 1230, Springer, 1986, pp. 221-234]) has been adapted for the solution of index-one DAEs, and the resulting integrator (SYMDAE) has been inserted into the multiple-shooting code (MSHDAE) previously developed by R. Lamour for differential-algebraic boundary-value problems. The standard version of MSHDAE uses a BDF integrator, which is not dichotomically stable, and for some stiff test problems this fails to integrate across the interval of interest, while the dichotomically stable integrator SYMDAE encounters no difficulty. Indeed, for such problems, the modified version of MSHDAE produces an accurate solution, and within limits imposed by computer word length, the efficiency of the solution process improves with increasing stiffness. For some nonstiff problems, the solution is also entirely satisfactory.

Consistent initialization for nonlinear index-2 differential-algebraic equation : large sparse systems in MATLAB

An important component of any initial-value solver for higher-index differential–algebraic equations consists in the computation of consistent initial values. In a recent paper [5], an algorithm is proposed which is applicable to a very general class of index-2 systems. Unfortunately, the computational expense is rather high. We present a modification of this approach, which gives rise to a MATLAB implementation capable of handling systems of moderate dimension (several thousands of unknowns). The algorithm is illustrated by examples.

Diagnosis of singular points of properly stated DAEs using automatic differentiation

The reliability of numerical results provided by conventional numerical integration methods for ODEs or DAEs depends on the properties of the problem. In particular, since the solution may not be unique at singular points, arbitrary solutions may be obtained. For DAEs, such singularities may occur, if the structure or the dimension of the spaces related to the DAE change. Moreover, even though a numerical singularity is not given in a strict mathematical sense, the numerical behavior may be analogous for sensitive problems. In this contribution, we aim at a characterization of this sensitivity, considering the condition number of a suitable matrix related to the DAE that is constructed using automatic differentiation. We show how this approach, which builds on the projector based analysis, can be applied to properly stated DAEs of index up to 2.

Stability of periodic solutions of index-2 differential algebraic systems

This paper deals with periodic index-2 differential algebraic equations and the question whether a periodic solution is stable in the sense of Lyapunov. As the main result, a stability criterion is proved. This criterion is formulated in terms of the original data so that it may be used in practical computations.

A new projector based decoupling of linear DAEs for monitoring singularities

For higher index differential-algebraic equations (DAEs) some components of the solution depend on derivatives of the right-hand side. In this context, two main results are pointed out here. On the one hand, a description of the different types of undifferentiated components involved in the DAE is obtained by a projector-based decoupling. To this end, we define a new decoupling based on the number of inherent differentiations of the right-hand side that are required to determine each component. On the other hand, we introduce characteristic values that characterize the robustness of our numerically determined index-classification and decoupling as well as a meaningful indicator that permit the diagnosis of singular points.