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Consistent Initialization and Perturbation Analysis for Abstract Differential-Algebraic Equations
Please always quote using this URN:urn:nbn:de:0296-matheon-3351
- 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.