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Consistent Initialization and Perturbation Analysis for Abstract Differential-Algebraic Equations
(2006)
In this paper we consider linear and time-invariant
differential-algebraic equations (DAEs) $E\dot{x}(t)=Ax(t)+f(t)$,
$x(0)=x_0$, where $x(\cdot)$ and $f(\cdot)$ are functions with
values in separable Hilbert spaces $X$ and $Z$. $E:X\To Z$ is
assumed to be a bounded operator, whereas $A$ is closed and defined
on some dense subspace $D(A)$ which is in general a proper subset of
$X$. Based on a decoupling of the algebraic and the differential
part, the set of initial values being consistent with the given
inhomogeneity will be parameterized. As a consequence of these results, we
will derive estimates for the trajectory $x(\cdot)$ in dependence of the initial
state $x_0$ and the inhomogeneity $f(\cdot)$. In the theory of
differential-algebraic equations, this is commonly known as
perturbation analysis.