Index concepts for linear differential-algebraic equations in finite and infinite dimensions
Mehmet Erbay, Birgit Jacob, Kirsten Morris, Timo Reis, Caren Tischendorf
TL;DR
The paper investigates index concepts for linear differential-algebraic equations on Banach spaces, defining six indices—resolvent, chain, radiality, nilpotency, differentiation, and perturbation—and establishing key comparisons: $p_{\rm rad}^{(E,A)}+1 \ge p_{\rm res}^{(E,A)} \ge p_{\rm nilp}^{(E,A)} = p_{\rm diff}^{(E,A)} = p_{\rm chain}^{(E,A)}$, with potential equality including $p_{\rm pert}^{(E,A)}$ when $A_1$ generates a $C_0$-semigroup. The analysis relies on the Weierstraß form and invariance under equivalence, showing that unlike finite-dimensional DAEs, indices may not all exist or be equivalent in infinite dimensions; under additional semigroup conditions, a stronger chain of equalities emerges. The work also introduces and situates each index (resolvent, chain, radiality, nilpotency, differentiation, perturbation) within a unified framework, relates them to well-posedness and Weierstraß theory, and provides examples illustrating possible disparities. Overall, the paper delineates what indices are meaningful in Banach-space DAEs, highlights their limitations, and outlines open questions about existence, equality, and practical relevance for infinite-dimensional systems.
Abstract
Different index concepts for linear differential-algebraic equations are defined in the general Banach space setting, and compared. For regular finite-dimensional linear differential-algebraic equations, all these indices exist and are equivalent. For infinite-dimensional systems, the situation is more complex. It is proven that although some indices imply others, in general they are not equivalent. The situation is illustrated with a number of examples.