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Topology of the orbit space of generalized linear systems. (English) Zbl 0741.93025
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.

93C05 Linear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations