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A flow-on-manifold formulation of differential-algebraic-equations

Baum, Ann-Kristin

Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin

We derive a flow formulation of differential-algebraic equations (DAEs), implicit differen-tial equations whose dynamics are restricted by algebraic constraints. Using the framework ofderivatives arrays and the strangeness-index, we identify the systems that are uniquely solv-able on a particular set of initial values and thus possess a flow, the mapping that uniquelyrelates a given initial value with the solution through this point. The flow allows to studysystem properties like invariant sets, stability, monotonicity or positivity. For DAEs, theflow further provides insights into the manifold onto which the system is bound to and intothe dynamics on this manifold. Using a projection approach to decouple the differential andalgebraic components, we give an explicit representation of the flow that is stated in theoriginal coordinate space. This concept allows to study DAEs whose dynamics are restrictedto special subsets in the variable space, like a cone or the nonnegative orthant.