Solvability conditions, consistency, and weak consistency for linear differential-algebraic equations and time-invariant singular systems: The general case

Abstract

We present several solvability concepts for linear differential-algebraic equations (DAEs) with constant coefficients on the positive time axis as well as for the associated singular systems, and investigate under which conditions these concepts are met. Next, we derive necessary and sufficient conditions for global consistency of initial conditions for the DAE as well as for the system, and generalize these conditions with respect to our concept of weak consistency. Our distributional approach enables us to generalize results in an earlier paper, where singular systems are assumed to have a regular pencil in the sense of Gantmacher. In particular, we establish that global weak consistency in the system sense is equivalent to impulse controllability.