On the Weierstraß form of infinite-dimensional differential algebraic equations (Q6617364)

The aim of this paper is to give sufficient conditions for an infinite-dimensional linear DAEs, to have a Weierstrass normal form. The DAE is defined using operators \(E, A\) on Hilbert spaces: \N\[\frac{d}{dt}Ex(t)=Ax(t)\tag{1}\] \N\(E\) is bounded and \(\ker E\) is non trivial, while \(A\) is a closed densily defined operator on its domain. The authors want to get results on the solvability for the class of port-Hamiltonian DAEs, used to model dissipative physical systems: \N\[\frac{d}{dt}Ex(t)=AQx(t)\tag{2}\] \Nwhere \(E, Q, A\) verify specific conditions (\(A\) is dissipative: see \url{https://en.wikipedia.org/wiki/Dissipative_operator}).\N\NThe first parts are devoted to the existence of solutions to \((1)\), with the introduction of the \textit{resolvent index}. The Weierstrass Normal Form existence is explained in the second part, using the \textit{radiality index} concept, related to a decomposition of the base space in a kind of Jordan form of the generalized matrix pencil associated to \((1)\). This leads to Theorems 3.3 and 3.9 on the existence of the Weierstrass form and its application to the system solvability (in infinite dimension) by the \textit{integrated semigroup} derived from the non-nilpotent part of operator \(A\). Parts 4. and 5. study the port-Hamiltonian DAEs \((2)\); its dissipativity is proved in Theorem 4.4: for any classical solution \(t \mapsto x(t)\) and \(t \ge 0\), \N\[\frac{d}{dt}(Ex(t), Qx(t)) \le 0.\] \N\NThe main propositions 5.1 and 5.2 conclude to the existence of the resolvent index, at most equal to \(3\), which is the generalization of results established in finite dimension, and the existence and uniqueness of solutions in this case - on the subspace where the initial conditions \(x(0)\) satisfy the algebraic constraints. This subspace is expressed as \N\[\operatorname{range}((\mu E-AQ)^{-1}E)^N,\] \Nthe range of a power of an operator related to \((2)\), \(N\) depending on the radiality index of \((2)\), \(\mu\) being outside the spectrum of \((E, A)\).\N\NThis paper is well-organized and demands standard knowledge of undergraduate linear algebra, interest for theories of DAEs in finite dimensions developed from 1990, while it is more difficult in terms of spectral analysis (for complex Hilbert spaces). However the presentation and the proofs are clear.