Differential algebraic system nodes
Mehmet Erbay, Birgit Jacob, Timo Reis
TL;DR
This work extends the theory of operator and system nodes to infinite-dimensional differential-algebraic systems with noninvertible $E$, formulating DAEs of the form $E\dot{x}(t)=Ax(t)+Bu(t)$, $y(t)=Cx(t)+Du(t)$ within an extrapolation-space framework. It constructs generalized extrapolation spaces, develops solution concepts (classical, mild, weak), and uses augmented Wong sequences to address solvability of inhomogeneous DAEs, culminating in the notion of differential-algebraic operator nodes (E-operator nodes) and their port-Hamiltonian extensions. The paper then integrates energy-based structure via port-Hamiltonian $E$-system nodes on quasi-Gelfand triples, deriving a dissipation inequality and establishing bounds on complex and real resolvent indices, with an illustrative PDE-based example showing index variability. Overall, the framework provides a rigorous, transfer-function-based approach to analyze and model infinite-dimensional port-Hamiltonian DAEs with algebraic constraints, enabling robust well-posedness and energy-dissipation analysis in multi-physics and boundary-control contexts.
Abstract
Infinite-dimensional differential algebraic equations (short DAEs) with input and output are studied. The concepts of operator nodes and system nodes are extended to systems which additionally may include algebraic constraints. Extrapolation spaces are investigated for differential-algebraic equations, and solutions of the extrapolated DAE are characterized using augmented Wong sequences. The resulting theory is then applied to characterize infinite-dimensional port-Hamiltonian DAEs.