Sylvester matrix equations play a prominent role in various areas such as control theory, medical imaging acquisition systems, model reduction, and stochastic control. Considering any uncertainty problems such as conflicting requirements during system process, instability of environmental conditions, distraction of any elements and noise, all for which the classical matrix equation is sometimes ill-equipped, fuzzy numbers represent the most effective tool that can be used to model matrix equations in the form of fuzzy equations. In most of the previous literature, the solutions of fuzzy systems are only presented with triangular fuzzy numbers. In this paper, we discuss fully fuzzy Sylvester matrix equation with positive and negative trapezoidal fuzzy numbers. An analytical approach for solving a fully fuzzy Sylvester matrix equation is proposed by transforming the fully fuzzy matrix equation into a system of four crisp Sylvester linear matrix equation. In obtaining the solution the Kronecker product and Vec-operator are used. Numerical examples are solved to illustrate the proposed method.

Sylvester矩阵方程式在各个领域都发挥着重要作用，例如控制理论，医学影像采集系统，模型简化和随机控制。考虑到任何不确定性问题，例如系统过程中的需求冲突，环境条件的不稳定性，任何元素的分散和噪声，所有这些问题有时装备不完善的经典矩阵方程式，模糊数代表可以用来建模的最有效工具模糊方程形式的矩阵方程。在以前的大多数文献中，模糊系统的解仅用三角模糊数表示。在本文中，我们讨论具有正负梯形模糊数的完全模糊Sylvester矩阵方程。通过将完全模糊矩阵方程转化为一个包含四个脆性Sylvester线性矩阵方程的系统，提出了一种求解完全模糊Sylvester矩阵方程的解析方法。为了获得解决方案，使用了Kronecker产品和Vec-operator。数值算例说明了所提出的方法。