On the solution of the generalized matrix Sylvester equations

Lyapunov matrix equations and their generalizations — linear matrix Sylvester equation widely used in the theory of stability of motion, control theory, as well as the solution of differential Riccati and Bernoulli equations, partial differential equations and signal processing. If the structure of the general solution of the homogeneous part of the Lyapunov equation is well studied, the solution of the inhomogeneous equation Sylvester and, in particular, the Lyapunov equation is quite cumbersome.
By using the theory of generalized inverse operators, A. A. Boichuk and S. A. Krivosheya establish a criterion of the solvability of the Lyapunov-type matrix equations $AX-XB=D$ and $X-AXB=D$ and investigate the structure of the set of their solutions. The article A. A. Boichuk and S. A. Krivosheya based on pseudo-inverse linear matrix operator $L$, corresponding to the homogeneous part of the Lyapunov type equation.
The article suggests the solvability conditions, as well as a scheme for constructing a particular solution of the inhomogeneous generalized equation Sylvester based on pseudo-inverse linear matrix operator corresponding to the homogeneous part of the linear matrix generalized Sylvester equation.
Using the technique of Moore–Penrose pseudo inverse matrices, we suggest an algorithm for finding a family of linearly independent solutions of the inhomogeneous generalized equation Sylvester and, in particular, the Lyapunov equation in general case when the linear matrix operator $L$, corresponding to the homogeneous part of the linear generalized matrix Sylvester equation, has no inverse.
We find an expression for family of linearly independent solutions of the inhomogeneous generalized equation Sylvester and, in particular, the Lyapunov equation in terms of projectors and Moore–Penrose pseudo inverse matrices. This result is a generalization of the result article A. A. Boichuk and S. A. Krivosheya to the case of linear generalized matrix Sylvester equation.
The suggested the solvability conditions and formula for constructing a particular solution of the inhomogeneous generalized equation Sylvester is illustrated by an examples.
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