Preprint 20-2013

Positivity characterization of nonlinear DAEs. Part I: A flow formular for linear and nonlinear DAEs using projections

Author(s) : Ann-Kristin Baum , Volker Mehrmann

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 20-2013

MSC 2000

34A09 Implicit equations, differential-algebraic equations

37C10 Vector fields, flows, ordinary differential equations

Abstract :
We construct an explicit solution formula of differential-algebraic equations (DAEs) that generalizes the concept of the flow to linear and nonlinear problems. Using the framework of the strangeness-index, we pursue a projection approach to separate the differential and algebraic components and remodel a given DAE as a semi-explicit system. Using the results on sub equations filtered out by projections that we prepared in [1] for differential and algebraic systems, we compute an explicit solution formula of the decomposed DAE. Verifying the unique relation between consistent initial values and the associated solution as well as similar functional properties as for the flow of a purely dynamical system, we define the flow associated with a DAE with regular strangeness-index. For linear problems, the flow is globally defined on every interval on which the coefficients satisfy certain constant rank assumptions. For general nonlinear problems, the flow is a local quantity that is defined in the neighborhood of every consistent value. The flow readily provides the tools to study immanent system properties like stability or contractivity. Constructed via projections, the flow is stated in the original variables and thus allows to study coordinate depending properties like positivity, in particular. [1] Positivity characterization of nonlinear DAEs. Part I: Decomposition of differential and algebraic equations using projections. A.K. Baum und V. Mehrmann. Preprint 18-2013. TU Berlin.

Keywords : differential-algebraic equations, flow, ordinary differential equations, projections