On inconsistent initial conditions for linear time-invariant differential-algebraic equations

Given an arbitrary initial value x₀^- for the differential-algebraic equation Ax˙(t)+Bx(t)=f(t), an initial value x₀⁺ can be selected from among all consistent initial values for that equation by means of the Laplace transform. We show that this choice is the only one that fulfils some simple, physically reasonable assumptions on linear systems´ behavior, thereby ruling out other values of x₀⁺ proposed in the literature. Our derivation is elementary compared to previous justifications of the above Laplace transform based method: We also characterize x₀⁺ by means of a system of linear equations involving A, B, derivatives of f, and x₀^-, which gives a new method to numerically calculate x₀⁺.