Solution to Sylvester equation associated to linear descriptor systems

This paper presents the solution of the constrained Sylvester equation associated to linear descriptor systems. This problem has been recently studied in Castelan and da Silva [On the solution of a Sylvester equation appearing in descriptor systems control theory, Systems Control Lett. 54 (2005) 109–117], where sufficient conditions for the existence of the solution are given. In the present paper, a simple and direct method is developed to solve this problem. This method shows that the conditions given in Castelan and da Silva [On the solution of a Sylvester equation appearing in descriptor systems control theory, Systems Control Lett. 54 (2005) 109–117] are necessary and sufficient.