Consistent initialization and perturbation analysis for abstract differential-algebraic equations

In this paper, we consider linear and time-invariant differential-algebraic equations (DAEs) Eẋ(t) = Ax(t) + f(t), x(0) = x ₀, where x(·) and f(·) are functions with values in Hilbert spaces X and Z. is assumed to be a bounded operator, whereas A is closed and defined on some dense subspace D(A). A transformation to a decoupling form leads to a DAE including an abstract boundary control system. Methods of infinite-dimensional linear systems theory can then be used to formulate sufficient criteria for an initial value being consistent with the given inhomogeneity. We will further derive estimates for the trajectory x(·) in dependence of the initial state x ₀ and the inhomogeneity f(·). In the theory of differential-algebraic equations, this is commonly known as perturbation analysis.