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

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, in 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.