A note on the controllability of pairs of matrices

Glória Cravo

doi:10.1016/j.crma.2014.06.007

Algebra

A note on the controllability of pairs of matrices

Presented by: the Editorial Board

Glória Cravo ¹

¹ Centro de Ciências Exactas e da Engenharia, Universidade da Madeira, 9020-105 Funchal, Madeira, Portugal

Comptes Rendus. Mathématique, Volume 352 (2014) no. 7-8, pp. 563-566.

Abstract
Résumé

Let F be an arbitrary field and let $n, p_{1}, p_{2}, p_{3}$ be positive integers such that $n = p_{1} + p_{2} + p_{3}$ . Let

(C_{1}, C_{2}) = ([\begin{matrix} C_{1, 1} & C_{1, 2} \\ C_{2, 1} & C_{2, 2} \end{matrix}], [\begin{matrix} C_{1, 3} \\ C_{2, 3} \end{matrix}]),

where the blocks

C_{i, j}

are of type

p_{i} \times p_{j}

,

i \in {1, 2}

,

j \in {1, 2, 3}

. Suppose that

C_{1, 1}

,

C_{1, 2}

and

C_{2, 1}

are known. In this note we identify sufficient conditions under which there exist

C_{1, 3}

,

C_{2, 2}

,

C_{2, 3}

such that the pair

(C_{1}, C_{2})

is completely controllable when

[C_{1, 1}^{⊤} {| C_{2, 1}^{⊤}]}^{⊤}

has full rank.

Soit F un corps arbitraire et soient $n, p_{1}, p_{2}, p_{3}$ des entiers positifs tels que $n = p_{1} + p_{2} + p_{3}$ . Soit

(C_{1}, C_{2}) = ([\begin{matrix} C_{1, 1} & C_{1, 2} \\ C_{2, 1} & C_{2, 2} \end{matrix}], [\begin{matrix} C_{1, 3} \\ C_{2, 3} \end{matrix}]),

où les blocs

C_{i, j}

sont de type

p_{i} \times p_{j}

,

i \in {1, 2}

,

j \in {1, 2, 3}

. On suppose les blocs

C_{1, 1}, C_{1, 2}, C_{2, 1}

connus. Nous identifions dans cette note des conditions suffisantes sous lesquelles il existe des blocs

C_{1, 1}, C_{1, 2}, C_{2, 1}

tels que la paire

(C_{1}, C_{2})

est complètement contrôlable quand le rang

[C_{1, 1}^{⊤} {| C_{2, 1}^{⊤}]}^{⊤} = p_{1}

.

Received: 2014-06-04
Accepted: 2014-06-16
Published online: 2014-07-01

DOI: 10.1016/j.crma.2014.06.007

Author's affiliations:

Glória Cravo ¹

¹ Centro de Ciências Exactas e da Engenharia, Universidade da Madeira, 9020-105 Funchal, Madeira, Portugal

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     author = {Gl\'oria Cravo},
     title = {A note on the controllability of pairs of matrices},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {563--566},
     publisher = {Elsevier},
     volume = {352},
     number = {7-8},
     year = {2014},
     doi = {10.1016/j.crma.2014.06.007},
     language = {en},
}

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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/

References
Cited by

[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.

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