Controllability in the max-algebra
Kybernetika, Tome 35 (1999) no. 1, p. [13].

Voir la notice de l'article dans Czech Digital Mathematics Library

We are interested here in the reachability and controllability problems for DEDS in the max-algebra. Contrary to the situation in linear systems theory, where controllability (resp observability) refers to a (linear) subspace, these properties are essentially discrete in the $\max $-linear dynamic system. We show that these problems, which consist in solving a $\max $-linear equation lead to an eigenvector problem in the $\min $-algebra. More precisely, we show that, given a $\max $-linear system, then, for every natural number $k\ge 1\,$, there is a matrix $\Gamma _k$ whose $\min $-eigenspace associated with the eigenvalue $1$ (or $\min $-fixed points set) contains all the states which are reachable in $k$ steps. This means in particular that if a state is not in this eigenspace, then it is not controllable. Also, we give an indirect characterization of $\Gamma _k$ for the condition to be sufficient. A similar result also holds by duality on the observability side.
Classification : 15A80, 93B05, 93B18, 93C65
Mots-clés : reachability; controllability; max-algebra 
@article{KYB_1999__35_1_a2,
     author = {Prou, Jean-Michel and Wagneur, Edouard},
     title = {Controllability in the max-algebra},
     journal = {Kybernetika},
     pages = {[13]},
     publisher = {mathdoc},
     volume = {35},
     number = {1},
     year = {1999},
     mrnumber = {1705527},
     zbl = {1274.93036},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/item/KYB_1999__35_1_a2/}
}
TY  - JOUR
AU  - Prou, Jean-Michel
AU  - Wagneur, Edouard
TI  - Controllability in the max-algebra
JO  - Kybernetika
PY  - 1999
SP  - [13]
VL  - 35
IS  - 1
PB  - mathdoc
UR  - https://geodesic-test.mathdoc.fr/item/KYB_1999__35_1_a2/
LA  - en
ID  - KYB_1999__35_1_a2
ER  - 
%0 Journal Article
%A Prou, Jean-Michel
%A Wagneur, Edouard
%T Controllability in the max-algebra
%J Kybernetika
%D 1999
%P [13]
%V 35
%N 1
%I mathdoc
%U https://geodesic-test.mathdoc.fr/item/KYB_1999__35_1_a2/
%G en
%F KYB_1999__35_1_a2
Prou, Jean-Michel; Wagneur, Edouard. Controllability in the max-algebra. Kybernetika, Tome 35 (1999) no. 1, p. [13]. https://geodesic-test.mathdoc.fr/item/KYB_1999__35_1_a2/