Extending Fully Fuzzy Sylvester Matrix Equations to Neutrosophic Environments: An Improved Solution Method

Abstract

The classical Sylvester matrix equations play a fundamental role in system analysis, model transformation and controller design. However, there are limits in handling uncertainty information as all parameters of the system are assumed to be precisely known. Uncertainty can occur in many real world applications so the classical mathematical models are insufficient to represent the system behaviour accurately. Even fuzzy Sylvester equations are unable to capture indeterminacy or contradiction within the data. Therefore, the study of solution framework for Fully Fuzzy Neutrosophic Sylvester Matrix Equation (FFNSME) that incorporates truth, indeterminacy and falsity components is introduced to overcome these limitations. Left– right triangular neutrosophic fuzzy numbers are used to represent uncertain parameters within the matrices. The Associated Linear Systems approach and the score function method are applied for deneutrosophication to enable computational processing while preserving embedded uncertainty. The findings demonstrate that the FFNSME framework provides a flexible representation of uncertainties compared to classical or fuzzy Sylvester equations. This enhances the reliability of system modeling, controller synthesis and matrix equation solutions when working with vague, contradictory or incomplete information. The formulation is applicable to a wide range of linear and nonlinear control problems. Future research may explore optimized numerical algorithms and extensions to large-scale, highdimensional and time-varying systems to further improve computational efficiency.