On Differential-Algebraic Equations with Bounded Spectrum in Banach Spaces

TL;DR

This work develops a framework to study homogeneous DAEs $\tfrac{d}{dt}Ex = Ax$ in Banach spaces with regular pencils, identifying when infinite-dimensional Wong-type decompositions exist under a bounded spectrum and polynomial resolvent growth, yielding a finite-index decoupling into an ODE plus a DAE of the form $\tfrac{d}{dt}Tx = x$ with a $T$ that is quasi-nilpotent. When the index is infinite, the DAE reduces to the quasi-nilpotent case, and the paper provides detailed criteria for the existence of $L^\infty$-solutions on $[0,\infty)$, $L^2$-solutions on compact intervals, and analytic solutions, illustrating that quasi-nilpotent dynamics can produce both trivial and nontrivial solution behavior. The approach relies on operator reductions via resolvents, Riesz-Dunford spectral projections, and spectral-projection techniques for pencils, connecting spectral properties to solvability. The results extend classical finite-dimensional DAE theory to infinite-dimensional Banach spaces, offering precise conditions for when a problem behaves like a decoupled ODE or exhibits richer quasi-nilpotent dynamics with specific solution classes and implications for stability and regularity.

Abstract

The Weierstraß form for regular DAEs in finite dimensions decouples a linear DAE into an ODE and the nilpotent part of the underlying pencil. Here, we provide necessary and sufficient conditions for the possibility of such a decomposition in the case of DAEs in Banach spaces. Moreover, we consider the larger class of linear operator pencils with bounded spectra and show that the associated homogeneous DAE can be reduced to an ODE and a seemingly simple DAE of the form $\frac d{dt}Tx = x$ with a quasi-nilpotent operator $T$. As examples show, there are cases with only the trivial solution and others with non-trivial solutions. We characterize the existence of $L^\infty$-solutions on the half-axis, $L^2$-solutions on compact time intervals, and analytic solutions.

Theorems & Definitions (40)

• Example 2.1
• Lemma 2.2
• Proposition 2.3
• Definition 3.1
• Lemma 3.2
• Theorem 3.3: m
• Proposition 3.4
• proof
• Lemma 3.5
• proof