Fully fuzzy Sylvester matrix equation

Abstract

The Sylvester equation arises in many application areas, for instance process and system control, and in the fuzzy setting, solution of this equation has been considered only in the case when the right-hand side matrix is a fuzzy matrix. This paper introduces the fully fuzzy Sylvester matrix equation A ˜ X ˜ - X ˜ B ˜ = C ˜, where the fuzzy matrices A ˜, B ˜ and C ˜ are of order n, m and n × m, respectively. The fuzzy matrix X ˜ is the sought after solution. A two-step scheme is developed for the solution of this system. The first step solves the 1-cut of the problem and the second step assigns unknown symmetric spreads to each row of the 1-cut expansions. The symmetric spreads are obtained by solving 2mn linear equations, depending on whether the coefficient matrices are both positive, or both negative, or a mixture of positive and negative. We show that the spreads are equivalent to zero when the membership function is one for both the coefficient matrices and the right-hand side matrix. Examples are given to illustrate the merit of the proposed procedure for determining a bounded solution to the fully fuzzy linear Sylvester equation.