Connection between the generalized Riccati equations and the standard reduced-order Riccati equations

In this paper, we consider the solution of a generalized differential Ricca ti equation and a generalized algebraic Riccati equation, which arise, resp ectively, from finite- and infinite-horizon linear-quadratic optimal contro l problems of descriptor systems or singular systems. Under the weakest ass umption, necessary and sufficient conditions for the existence of a solutio n to the generalized Riccati equations are obtained in terms of standard re duced-order Riccati equations. Furthermore, sufficient conditions for the e xistence of a generalized positive semidefinite solution of the generalized differential Riccati equation, and a strongly stabilizing solution of the generalized algebraic Riccati equation are also studied.