finding a generalized positive solution equation for a trapezoidal fully fuzzy sylvester matrix

The determination of the solvability of sylvester matrix equations is crucial in various domains of control theory and systems theory. in numerous applications, employing fuzzy numbers is more suitable for representing certain system parameters compared to using exact values. previous studies predominantly focus on solutions that incorporate triangular fuzzy numbers for the fuzzy sylvester matrix problem. this work presents two analytical methods aimed at addressing the positive trapezoidal fully fuzzy sylvester matrix equation. by utilizing contemporary fuzzy arithmetic multiplication operations, we reformulate this equation into a corresponding system of crisp sylvester matrix equations. the investigation covers the necessary and sufficient conditions for the existence of positive fuzzy solutions and their uniqueness for the positive trapezoidal fully fuzzy sylvester matrix equation. moreover, the relationship between the positive trapezoidal fully fuzzy sylvester matrix equation and the system of sylvester matrix equations is explored. an illustrative example is provided to demonstrate the effectiveness of the proposed approaches.