Simultaneously controllable fuzzy matrices. (English) Zbl 1100.15008
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 K. H. Kim and F. W. Roush [Fuzzy Sets Syst. 4, 293–315 (1980; Zbl 0451.20055)] and examined by H. Hashimoto [Fuzzy Sets Syst. 11, 157–162 (1983; Zbl 0523.15013)], X. T. Peng [Fuzzy Sets Syst. 19, 47–50 (1986; Zbl 0606.15008)], W. Kołodziejczyk [Fuzzy Sets Syst. 22, 297–302 (1987; Zbl 0623.15019)] and Ch. Hao [Fuzzy Sets Syst. 45, 219–222 (1992; Zbl 0752.15017)]. A characterization of such matrices was obtained by 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 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 S. M. Guu, H.-H. Chen and C.-T. Pang [Fuzzy Sets Syst. 121, 203–207 (2001; Zbl 0994.15017)].
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 S. M. Guu, H.-H. Chen and C.-T. Pang [Fuzzy Sets Syst. 121, 203–207 (2001; Zbl 0994.15017)].
Reviewer: Józef Drewniak (Rzeszów)
MSC:
| 15B33 | Matrices over special rings (quaternions, finite fields, etc.) |
| 15A30 | Algebraic systems of matrices |
| 08A72 | Fuzzy algebraic structures |
| 65F30 | Other matrix algorithms (MSC2010) |
Keywords:
fuzzy matrix; controllable matrix; nilpotent matrix; mixed relation powers; convergence of powers; oscillation of powers; algorithm for checking of controllabilityCitations:
Zbl 0451.20055; Zbl 0523.15013; Zbl 0606.15008; Zbl 0623.15019; Zbl 0752.15017; Zbl 0749.15014; Zbl 1028.15015; Zbl 0994.15017References:
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