Closed-form solutions to the nonhomogeneous Yakubovich-transpose matrix equation

The nonhomogeneous Yakubovich-transpose matrix equation X−AXTB=CY+RX−AXTB=CY+R, which contains the well-known Kalman–Yakubovich-transpose matrix equation and general discrete Lyapunov-transpose matrix equation as special cases, has many important applications in control system theory. This study presents two methods to obtain the closed-form solutions of the nonhomogeneous Yakubovich-transpose matrix equation. Moreover, the equivalent forms of the solutions are provided and one of the solutions is established with the controllability matrix, the observability matrix and symmetric operator matrix.