On the structure of the solutions of discrete-time algebraic Riccati Equation with singular closed-loop matrix

The classical discrete-time algebraic Riccati equation (DARE) is considered in the case when the corresponding closed-loop matrix is singular. It is shown that in this case all the symmetric solutions of the DARE coincide along some directions. A parametrization of the set of solutions in terms of the reduced-order DARE is then obtained. This parametrization provides an algorithm (that appears to be computationally very attractive when the multiplicity of the zero eigenvalue of the closed-loop matrix is large) for the computation of the solutions of the DARE. The same issue for the generalized DARE is also addressed.