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Matrix pencil characterization of almost (A,B)-invariant subspaces: A classification of geometric concepts. (English) Zbl 0552.93018

The equivalence between the algebraic, matrix pencil, characterization of the subspaces of the ’extended’ geometric theory and their dynamic characterization is established. As a result, a complete classification of almost (A,B)-invariant, (A,B)-invariant, almost controllability and controllability subspaces is derived in terms of matrix pencil invariants. The frequency propagation aspects of infinite spectrum (A,B)- invariant subspaces are investigated and it is shown that they are limits of closed-loop eigenspaces with arbitrarily large eigenvalues. Finally, the importance of the infinite frequency subspaces in the study of the asymptotic behaviour of the closed-loop eigenspaces and eigenvalues under scalar gain output feedback is discussed.

MSC:

93B25 Algebraic methods
15A18 Eigenvalues, singular values, and eigenvectors
47A15 Invariant subspaces of linear operators
15A21 Canonical forms, reductions, classification
93B05 Controllability
93B10 Canonical structure
93C05 Linear systems in control theory
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