Solutions to the linear transpose matrix equations and their application in control

Abstract

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.