Solvability, controllability, and observability of continuous descriptor systems

In this paper, we investigate the properties of the continuous descriptor system
E\dot{x}(t)=Ax(t)+Bu(t), 0\leqt\leqb
where
E, A,
and
B
are complex and possibly singular matrices and
u(t)
is a complex function differentiable sufficiently many times. The traditional approach to such systems is to separate the state equations from the algebraic equations. However, such algorithms usually destroy the natural, physically-based sparsity and structure of the original system. Therefore, we consider descriptor systems in their original form. Such systems possess numerous properties not shared by the well-known state variable systems. First, we relate classical theories of matrix pencils to the solvability of descriptor systems. Then we extend the concepts of reachability, controllability, and observability of state variable systems to descriptor systems, and describe the set of reachable states for descriptor systems.