More geometry about singular systems

This paper presents a new way of introducing invariant subspaces for singular systems : this leads to somewhat known geometric algorithms but, in most cases, with different initial conditions and enlightens the accurate duality holding for singular systems. Many results from the proper case can thus be generalized : the algebraic characterization of these invariant subspaces (see SCHUMACHER [10]), the geometric definition for complete controllability indexes [15] (see KUČERA and ZAGALAK [14]), the MORSE's canonical decomposition [2]... Singular systems are systems described by : E {(t) = Ax(t) + Bu(t) { t>0 (1) y(t) = Cx(t) where x(t)∈ X = Rn, u(t) ∈ U = Rm, y(t)∈ y = RP, and RankE ≤ n. A lot of interest has been brought to the study of these systems (many references can be found for instance in [11]) and many authors have generalized well-known concepts and results of the proper case (E=I) to those more general systems. As concerns the geometric approach of proper systems, the main contributions are due to WONHAM & MORSE (see for instance [4]) and BASILE & MARRO [1] ... (for the notions of invariant subspaces) and to WlLLEMS [6] (for the notions of almost invariant subspaces). One can describe those particular subspaces of the state space within the algebraic framework proposed by SCHUMACHER [10] which is directly resting upon the study of trajectories. Denote B the image of B and K the kernel of C in (1) and recall that a subspace V is called (A,E,B) invariant whence [5], [11]: AV ⊂ V + B (2) with : v =: EV