Abstract
The fuzzy Sylvester matrix equationAX~+X~B=C~in whichA,Barem×mandn×ncrisp matrices, respectively, andC~is anm×nLR fuzzy numbers matrix is investigated. Based on the Kronecker product of matrices, we convert the fuzzy Sylvester matrix equation into an LR fuzzy linear system. Then we extend the fuzzy linear system into two systems of linear equations according to the arithmetic operations of LR fuzzy numbers. The fuzzy approximate solution of the original fuzzy matrix equation is obtained by solving the crisp linear systems. The existence condition of the LR fuzzy solution is also discussed. Some examples are given to illustrate the proposed method.
Highlights
System of simultaneous matrix equations plays a major role in various areas such as mathematics, physics, statistics, engineering, and social sciences
We extend the fuzzy LR linear system (16) into two systems of linear equations according to arithmetic operations of LR fuzzy numbers
In this work we presented a model for solving fuzzy Sylvester matrix equations AX + XB = Cwhere A and B are m × m and n × n crisp matrices, respectively, and Cis an m × n arbitrary LR fuzzy numbers matrix
Summary
System of simultaneous matrix equations plays a major role in various areas such as mathematics, physics, statistics, engineering, and social sciences. For a fuzzy linear matrix equation which always has a wide use in control theory and control engineering, few works have been done in the past decades. In 2009, Allahviranloo et al [21] studied the fuzzy linear matrix equation, (FLME) of the form AXB = C. In 2011, Guo and Gong [22, 23] investigated a class of simple fuzzy matrix equations AX = Bby the undetermined coefficients method and studied least squares solutions of the inconsistent fuzzy matrix equation AX = Bby using generalized inverses of matrices. In 2011, Guo [24] studied the approximate solution of fuzzy Sylvester matrix equations with triangular fuzzy numbers.
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