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

Some closed-form solutions are provided for the nonhomogeneous Yakubovich-conjugate matrix equation X-AX¯F=BY+R with X and Y being unknown matrices. The presented solutions can offer all the degrees of freedom which is represented by an arbitrarily chosen parameter matrix. The primary feature of the solutions is that the matrices F and R are not restricted to be in any canonical form, or may be even unknown a priori. One of the solutions is neatly expressed in terms of controllability matrices and observability matrices.