Regularizing generalized linear systems by means a derivative feedback

We consider generalized linear time invariant systems in the form Ex(t) = Ax(t) + Bu(t) with E, A ∈ M_n×m(C), B × M_n×m(C), arising naturally in a variety of circumstances fro example they are used in modelling of mechanical multibody system. Obviously, the regularity condition det E ≠ 0 guarantees the existence and uniqueness of solutions. In this work we study necessary and sufficient conditions in order that the matrix E can be regularized by means of a derivative feedback, that is to say conditions that ensure the existence if a control u(t) = -Vx˙(t) + w(t) with V ∈ M_n×m(C) such that the matrix E + BV in the closed loop system (E + BV)x˙(t) = Ax(t) + Bw(t) is regular and consequently the closed loop system is uniquely solvable. We also study conditions for controllability of the closed loop system ensuring the existence of feedback w(t) = Ux(t) + w(t) with V ∈ M_n×m(C), such that the system (E + BV)x˙(t) = (A + BU)x(t) + Bw(t) has a stable solution. Finally, a stratification of the space of orbits of the systems that can be regularized is presented.