In this paper, we study the solutions to the linear transpose matrix equations \(AX+X^{T}B=C\) and \(AX+X^{T}B=CY\), which have many important applications in control theory. By applying Kronecker map and Sylvester sum, we obtain some necessary and sufficient conditions for existence of solutions and the expressions of explicit solutions for the Sylvester transpose matrix equation \(AX+X^{T}B=C\). Our conditions only need to check the eigenvalue of \( B^{T}A^{-1}\), and, therefore, are simpler than those reported in the paper (Piao et al. in J Frankl Inst 344:1056–1062, 2007). The corresponding algorithms permit the coefficient matrix C to be any real matrix and remove the limit of \(C=C^{T}\) in Piao et al. Moreover, we present the solvability and the expressions of parametric solutions for the generalized Sylvester transpose matrix equation \(AX+X^{T}B=CY\) using an alternative approach. A numerical example is given to demonstrate that the introduced algorithm is much faster than the existing method in the paper (De Terán and Dopico in 434:44–67;2011). Finally, the continuous zeroing dynamics design of time-varying linear system is provided to show the effectiveness of our algorithm in control.

在本文中，我们研究了线性转置矩阵方程\（AX + X ^ {T} B = C \）和\（AX + X ^ {T} B = CY \）的解，它们在控制理论。通过应用Kronecker映射和Sylvester和，我们为Sylvester转置矩阵方程\（AX + X ^ {T} B = C \）的解的存在和显式解的表达式提供了一些充要条件。我们的条件只需要检查\（B ^ {T} A ^ {-1} \）的特征值，因此比本文报道的条件更简单（Piao等人在J Frankl Inst 344：1056– 1062，2007）。相应的算法允许系数矩阵C为任何实矩阵，并消除\（C = C ^ {T} \）的限制在Piao等。此外，我们使用另一种方法给出了广义Sylvester转置矩阵方程\（AX + X ^ {T} B = CY \）的可解性和参数解的表达式。数值例子表明，引入的算法比本文中的现有方法要快得多（DeTerán和Dopico in 434：44-67； 2011）。最后，提供了时变线性系统的连续调零动力学设计，以证明该算法在控制中的有效性。