Algebra
A note on the controllability of pairs of matrices
Comptes Rendus. Mathématique, Volume 352 (2014) no. 7-8, pp. 563-566.

Let F be an arbitrary field and let $n,p1,p2,p3$ be positive integers such that $n=p1+p2+p3$. Let

 $(C1,C2)=([C1,1C1,2C2,1C2,2],[C1,3C2,3]),$
where the blocks $Ci,j$ are of type $pi×pj$, $i∈{1,2}$, $j∈{1,2,3}$. Suppose that $C1,1$, $C1,2$ and $C2,1$ are known. In this note we identify sufficient conditions under which there exist $C1,3$, $C2,2$, $C2,3$ such that the pair $(C1,C2)$ is completely controllable when $[C1,1⊤|C2,1⊤]⊤$ has full rank.

Soit F un corps arbitraire et soient $n,p1,p2,p3$ des entiers positifs tels que $n=p1+p2+p3$. Soit

 $(C1,C2)=([C1,1C1,2C2,1C2,2],[C1,3C2,3]),$
où les blocs $Ci,j$ sont de type $pi×pj$, $i∈{1,2}$, $j∈{1,2,3}$. On suppose les blocs $C1,1,C1,2,C2,1$ connus. Nous identifions dans cette note des conditions suffisantes sous lesquelles il existe des blocs $C1,1,C1,2,C2,1$ tels que la paire $(C1,C2)$ est complètement contrôlable quand le rang $[C1,1⊤|C2,1⊤]⊤=p1$.

Accepted:
Published online:
DOI: 10.1016/j.crma.2014.06.007

Glória Cravo 1

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Glória Cravo. A note on the controllability of pairs of matrices. Comptes Rendus. Mathématique, Volume 352 (2014) no. 7-8, pp. 563-566. doi : 10.1016/j.crma.2014.06.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.06.007/

[1] G. Cravo On controllability of pairs of matrices, Linear Algebra Appl. (2014) (in preparation, to be submitted to)

[2] G. Cravo; J.A. Dias da Silva; F.C. Silva Characteristic polynomials and controllability of partially prescribed matrices, Linear Algebra Appl., Volume 335 (2001), pp. 157-166

[3] T. Kailath Linear Systems, Prentice-Hall, Englewood Cliffs, NJ, 1980

Cited by Sources:

This research was done within the activities of the Centro de Estruturas Lineares e Combinatórias.