Geodesic
On the question of a constructive controllability criterion. Pt~I. Cyclic invariant subspaces
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 19 (2023) no. 2, pp. 283-299

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The rank of the Kalman's controllability matrix of linear systems depends on the bases of the invariant cyclic subspaces of the state matrix generated by the columns of the input matrix. The case of the Jordan form of the state matrix and scalar control is studied in detail. It is shown that the dimension of cyclic subspaces is determined by the index numbers of the first non-zero elements of the coordinate blocks of the columns of the input matrix. The formation of the bases of these subspaces is completely disclosed. Based on this, the basis of the space of a constructive control system is constructed.
Keywords: controllability, system structure, cyclic invariant subspaces. 
@article{VSPUI_2023_19_2_a12,
     author = {E. A. Kalinina and A. M. Kamachkin and N. A. Stepenko and G. Sh. Tamasyan},
     title = {On the question of a constructive controllability criterion. {Pt~I.} {Cyclic} invariant subspaces},
     journal = {Vestnik Sankt-Peterburgskogo universiteta. Prikladna\^a matematika, informatika, processy upravleni\^a},
     pages = {283--299},
     publisher = {mathdoc},
     volume = {19},
     number = {2},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSPUI_2023_19_2_a12/}
}
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E. A. Kalinina; A. M. Kamachkin; N. A. Stepenko; G. Sh. Tamasyan. On the question of a constructive controllability criterion. Pt~I. Cyclic invariant subspaces. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 19 (2023) no. 2, pp. 283-299. http://geodesic.mathdoc.fr/item/VSPUI_2023_19_2_a12/