Spectral theory of infinite dimensional dissipative Hamiltonian systems
Christian Mehl, Volker Mehrmann, Michał Wojtylak
TL;DR
This work develops the spectral theory of operator pencils in infinite dimensions, focusing on singular pencils and their link to the existence and (non)uniqueness of solutions to $E\dot x(t)=Ax(t)$. It introduces five natural singularity notions, analyzes their relationships, and shows that finite-dimensional intuitions do not automatically extend to the infinite-dimensional setting, especially for general pencils. The authors then specialize to dissipative Hamiltonian pencils of the form $P(\lambda)=\lambda E-BQ$ with $Q^*E$ selfadjoint nonnegative and $B$ dissipative, obtaining a complete uniqueness characterization for the associated operator DAEs via energy-based arguments and spectral conditions. The results illuminate how structure in $E,B,Q$ yields decisive criteria for existence and uniqueness and have practical implications for energy-based modeling and numerical treatments of DAEs in physics and engineering.
Abstract
The spectral theory for operator pencils and operator differential-algebraic equations is studied. Special focus is laid on singular operator pencils and three different concepts of singularity of operator pencils are introduced. The concepts are analyzed in detail and examples are presented that illustrate the subtle differences. It is investigated how these concepts are related to uniqueness of the underlying algebraic-differential operator equation, showing that, in general, classical results known from the finite dimensional case of matrix pencils and differential-algebraic equations do not prevail. The results are then studied in the setting of structured operator pencils arising in dissipative differential-algebraic equations. Here, unlike to the general infinite-dimensional case, the uniqueness of solutions to dissipative differential-algebraic operator equations is closely related to the singularity of the pencil.