On nilpotent singular systems

We derive some results for linear differential-algebraic equations (DAEs) of the form N(t)z′(t)+z(t)=h(t), where N(t) is a smooth nilpotent matrix for all t concerned and such that the system is uniquely solvable, i.e., has exactly one solution for each smooth h. Such systems play a fundamental role in the investigation of more general DAEs, but their theory is still incomplete. We give some sufficient conditions for unique solvability, and a global representation for the solution operator constructed in terms of a finite set of special solutions.