A Flow-on-Manifold Formulation of Differential-Algebraic Equations

Abstract

Differential-algebraic equations (DAEs) are coupled systems of differential and algebraic equations arising in the modeling of dynamic processes that are restricted by auxiliary, algebraic constraints. Examples are, e.g., multibody systems, where the movement of the system is forced to a predefined trajectory by connected joints, electrical circuits or networks, where connections and loops lead to algebraic relations of the flow and effort variables, or biological systems, reaction networks or advection-diffusion equations, where the evolution of the process is restricted by balance equations and conservation laws. Modeling dynamic processes in economic or social sciences, in biological or chemical engineering, the analyzed values typically represent real-valued quantities like the amount of goods or individuals or the density of a chemical or biological species that cannot take negative values. In combination with the algebraic constraints, this property leads to the notion of positive DAEs, i.e., systems whose solutions remain componentwise nonnegative whenever the initial value is nonnegative.In this work, we generalize the concept of the flow from ordinary differential equations (ODEs) to DAEs and use this framework to study system properties like invariant sets and positivity. To decouple the differential and algebraic components in a DAE, we develop an approach that allows to parameterize embedded submanifolds using projections and that extends the idea of projections to embedded submanifold. Combined with the theory of the strangeness-index, we apply this projection approach to remodel a given DAE as a set of explicit differential and algebraic equations. Solving the decoupled system and showing existence and uniqueness of the solution representation, we generalize the notion of the flow to DAEs. For linear systems, we generalize Duhamel's formula. Based on the flow, we extend the results of invariant sets from ODEs to DAEs and we characterize linear and nonlinear DAEs with regard to positivity. ZUSAMMENFASSUNG

Linear-Quadratic Optimal Control of Differential-Algebraic Systems: The Infinite Time Horizon Problem with Zero Terminal State

“…Moreover, a generalization of the KYP inequality and Lur'e equation to differential-algebraic equations with time-varying coefficients seems possible. For example, we believe that the system space turns to a time-varying manifold that could be treated by the theory elaborated in [5]. This could be even further extended to nonlinear differential-algebraic equations, but in this case, also general (non-quadratic) cost functionals are of interest.…”