Indefinite sign Riccati theory for descriptor systems

We present a general Riccati theory for systems in descriptor form considered under the weakest possible assumptions imposed on the coefficent matrices. A Riccati-like equation of descriptor form is introduced and its stabilizing solution is characterized in terms of the associated extended Hamiltonian pencil (EHP) whilst the computation of such a solution is reduced to solving a generalized eigenvalue problem for a singular EHP. The results exposed in the paper are generalizations to the game-theoretic (sign indefinite) and singular cases of the (positivity) Riccati theory developed by several authors in the framework of descriptor systems. Possible applications range from various nonstandard and J-spectral factorizations to H² and H^∞ control of descriptor systems.