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Top1. Introduction
Many real problems in control systems are related to the solvability of the Generalized Sylvester Matrix Equation (GSME) in the form 
, either using analytical or numerical methods. The GSME has important applications in the design and analysis of linear control systems (Datta, 2004), observer design (Tsui, 1988), reduction of large-scale dynamical systems (Van Dooren, 2000), restoration of noisy images (Bouhamidi & Jbilou, 2007; Calvetti & Reichel, 1996), medical imaging data acquisition and filtering (Bouhamidi & Jbilou, 2007). Researchers for many years have proposed many analytical and numerical methods for solving the GSME with crisp numbers (Dehghan & Shirilord, 2019; Ding et al., 2008; Sasaki & Chansangiam, 2020). However, in many applications, the classical GSME is not well equipped to handle uncertainty in real-life problems such as conflicting requirements during the system process, the distraction of any elements and noise. Therefore, the crisp numbers need to be replaced by fuzzy numbers. Lukasiewicz and Tarski (1920) studied the fuzzy logic as infinite valued logic, and Lotfi Zadeh (1965) introduced the fuzzy set theory and Fuzzy Relation Equations (FREs) with the max-min composition were introduced by Sanchez (1976). Solving FREs has become one of the most extensively studied problems in fuzzy sets and fuzzy logic (De Baets, 2000) (di Nola et al., 1989). In addition to the FREs, many researchers considered different types of fuzzy intervals (Yu et al., 2021)(Li et al., 2020) (Li & Liu, 2014) and different types of fuzzy numbers such as Triangular Fuzzy Numbers (TFNs) (Liang & Li, 2019) (Ye & Li, 2020), and Trapezoidal Fuzzy Numbers (TrFNs) (Vijayalakshmi & Sattanathan, 2011)(Bansal, 2011; Vahidi & Rezvani, 2013).
When all parameters of the GSME are in fuzzy form, it is called a Generalized Fully Fuzzy Sylvester Matrix Equation (GFFSME).
where,
,
,
,
,
and
is called GFFSME.