Singular Riccati theory via extended symplectic pencils

A Riccati like equation, termed as the generalized (discrete-time) algebraic Riccati equation, which incorporates as special cases both the standard and the constrained discrete-time algebraic Riccati equations is introduced and investigated under the weakest possible assumptions imposed on the initial data. A complete characterization of the conditions under which such an equation of general form has a stabilizing solution is presented in terms of the so called proper deflating subspace of the extended symplectic pencil. An evaluation of an associated quadratic index along constrained stable trajectories is given in terms of the stabilizing solution to the generalized Riccati equation. Possible applications of the developed theory range from nonstandard spectral and inner-outer factorizations to singular H² and H^∞ control. The results exposed in the present paper could be seen as an extension to singular cases of the discrete-time algebraic Riccati equation theory of indefinite sign