# Positivity characterization of nonlinear DAEs. Part I: Decomposition of differential and algebraic equations using projections

 dc.contributor.author Baum, Ann-Kristin dc.contributor.author Mehrmann, Volker dc.date.accessioned 2021-12-17T10:11:20Z dc.date.available 2021-12-17T10:11:20Z dc.date.issued 2013-07-18 dc.description.abstract In this paper, we prepare the analysis of differential-algebraic equations (DAEs) with regard to properties as positivity, stability or contractivity. To study these properties, the differential and algebraic components of a DAE must be separated to quantify when they exhibit the desired property. For the differential components, the common results for ordinary differential equations (ODEs) can be extended, whereas the algebraic components have to satisfy certain boundedness conditions. In contrast to stability or contractivity, for the positivity analysis, the system cannot be decomposed by changing the variables as this also changes the coordinate system in which we want to study positivity. Therefore, we consider a projection approach that allows to identify and separate the differential and algebraic components while preserving the coordinates. In Part I of our work, we develop the decomposition by projections for differential and algebraic equations to prepare the analysis of DAEs in Part II. We explain how algebraic and differential equations are decomposed using projections and discuss when these decompositions can be decoupled into independent sub components. We analyze the solvability of these sub components and study how the decomposition is reflected in the solution of the overall system. For algebraic equations, this includes a relaxed version of the implicit function theorem in terms of projections allowing to characterize the solvability of an algebraic equation in a subspace without actually filtering out the regular components by changing the variables. In Part II, we use these results and the decomposition approach to decompose DAEs into the differential and algebraic components. This way, we obtain a semi-explicit system and an explicit solution formula in the original coordinates that we can study with regard to properties as positivity, stability or contractivity. en dc.identifier.issn 2197-8085 dc.identifier.uri https://depositonce.tu-berlin.de/handle/11303/15778 dc.identifier.uri http://dx.doi.org/10.14279/depositonce-14551 dc.language.iso en en dc.rights.uri http://rightsstatements.org/vocab/InC/1.0/ en dc.subject.ddc 510 Mathematik en dc.subject.other differential-algebraic equations en dc.subject.other ordinary differential equations en dc.subject.other algebraic equations en dc.subject.other projections en dc.subject.other invariant subspace en dc.title Positivity characterization of nonlinear DAEs. Part I: Decomposition of differential and algebraic equations using projections en dc.type Research Paper en dc.type.version submittedVersion en tub.accessrights.dnb free en tub.affiliation Fak. 2 Mathematik und Naturwissenschaften::Inst. Mathematik de tub.affiliation.faculty Fak. 2 Mathematik und Naturwissenschaften de tub.affiliation.institute Inst. Mathematik de tub.publisher.universityorinstitution Technische UniversitÃ¤t Berlin en tub.series.issuenumber 2013, 18 en tub.series.name Preprint-Reihe des Instituts fÃ¼r Mathematik, Technische UniversitÃ¤t Berlin en tub.subject.msc2000 34A09 Implicit equations, differential-algebraic equations en tub.subject.msc2000 47A15 Invariant subspaces en tub.subject.msc2000 26B10 Implicit function theorems, Jacobians, transformations with several variables en

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