Simultaneously controllable fuzzy matrices
The paper deals with multiple max-min products of matrices over the lattice ([0,1],max,min) (fuzzy matrices). The canonical form of fuzzy matrices was proposed by \textit{K. H. Kim} and \textit{F. W. Roush} [Fuzzy Sets Syst. 4, 293--315 (1980; Zbl 0451.20055)] and examined by \textit{H. Hashimoto} [Fuzzy Sets Syst. 11, 157--162 (1983; Zbl 0523.15013)], \textit{X. T. Peng} [Fuzzy Sets Syst. 19, 47--50 (1986; Zbl 0606.15008)], \textit{W. Kołodziejczyk} [Fuzzy Sets Syst. 22, 297--302 (1987; Zbl 0623.15019)] and \textit{Ch. Hao} [Fuzzy Sets Syst. 45, 219--222 (1992; Zbl 0752.15017)]. A characterization of such matrices was obtained by \textit{J. X. Li} [Fuzzy Sets Syst. 45, 313--319 (1992; Zbl 0749.15014)] under the new name `controllable fuzzy matrices'. As a generalization we have the notion of simultaneous controllable matrices by \textit{Y. Y. Lur} et al. [Linear Algebra Appl. 367, 37--45 (2003; Zbl 1028.15015)]. In this paper the results of Li [loc. cit.] are extended for a family of simultaneously controllable matrices. In particular we have a characterization by the use of nilpotent matrices and an algorithm for checking of simultaneous controllability. These results are used in examination of convergence of mixed products of a family of fuzzy matrices according to the definition by \textit{S. M. Guu, H.-H. Chen} and \textit{C.-T. Pang} [Fuzzy Sets Syst. 121, 203--207 (2001; Zbl 0994.15017)].
algorithm for checking of controllability
scientific article; zbMATH DE number 5003151
scientific article