Regularization of linear descriptor systems with variable coefficients

We study linear descriptor control systems with rectangular variable coefficient matrices. We introduce condensed forms for such systems under equivalence transformations and use these forms to detect whether the system can be transformed to a uniquely solvable closed loop system via state or derivative feedback. We show that under some mild assumptions every such system consists of an underlying square subsystem that behaves essentially like a standard state space system, plus some solution components that are constrained to be zero.