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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.

##### MSC:
 93C05 Linear systems in control theory 93C15 Control/observation systems governed by ordinary differential equations
##### Keywords:
restricted systems equivalence; scaling action