Controllability in the max-algebra

We ale interested here in the reachability and controllability problems for DEDS in the max-algebra. Contrary to the situation in linear systems theor y, where controllability (resp observability) refers to a (linear) subspace , these properties are essentially discrete in the max-linear dynamic syste m. We show that these problems, which consist in solving a max-linear equat ion lead to an eigenvector problem in the min-algebra. More precisely, we s how that, given a max-linear system, then, for every natural number k great er than or equal to 1, there is a matrix Gamma(k) whose min-eigenspace asso ciated with the eigenvalue 1 (or min-fixed points set) contains all the sta tes which are reachable in k steps. This means in particular that if a stat e 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.