Existence and representation of stabilizing solutions to generalized algebraic Riccati equations

This paper investigates the existence and representation of stabilizing solutions to a class of generalized algebraic Riccati equation (GARE), which is often encountered in analysis and synthesis problems of continuous-time descriptor systems, such as H1 control and positive real control. The GARE under consideration consists of a couple of equations A^T X + X^T A + X^T RX + Q = 0 and E^T X = X^TE ¿ 0, where E, A, R, Q ¿ ¿^n×n are known and satisfy that R^T = R, Q^T = Q and E is singular. Based on the Hamiltonian matrix pencil technique, a necessary and sufficient condition for the existence of stabilizing solutions to the GARE is obtained, and a representation of all such solutions is given. As an intermediate step, a representation of all invertibility-producing solutions (that is, solutions X such that A + RX is invertible) to an algebraic Riccati equation like A^T X + X^T A + X^T RX + Q = 0 is derived. Furthermore, numerical algorithms corresponding to main conclusions are designed, which offer convenience for applications of these obtained conclusions. The effectiveness of these proposed approaches is shown by an example. These obtained conclusions can be viewed as a supplementary version of the existing results.