Topology of the orbit space of generalized linear systems (Q1176429)

Generalized linear systems \(Ex'=Ax+Bu\) which are admissible are studied under the `` restricted systems equivalence'' transformation induced by \((E,A,B)\to (MEN^{-1},MAN^{-1},MB)\) and the `` scaling action'' \((E,A,B)\to (aE+bA,cE+dA,B)\), \(ad-bc\neq 0\). The class of systems which are studied also assume ``controllability'' i.e. \([\lambda E-\mu A,B]\) of full rank for each \((\lambda,\mu)\neq (0,0)\). It is proved that the quotient space of the space of controllable systems under restricted systems equivalence is a smooth compact projective algebraic manifold and the homology groups are calculated.