Four extremal solutions of discrete-time algebraic Riccati equations: existence theorems and computation

Algebraic Riccati equations (AREs) have been extensively applied in linear optimal control problems and many efficient numerical methods were developed. The stabilizing (or almost stabilizing) solution that all eigenvalues of its closed-loop matrix are contained in the open (or closed) unit disk of the complex plane has attracted the most attention among all Hermitian solutions of the ARE in the past works. Nevertheless, it is an interesting and challenging issue in finding the extremal solutions of AREs which play an important role in the applications. The contribution of this paper is twofold. Firstly, the existence of these extremal solutions is established under the framework of fixed-point iteration. Secondly, an accelerated fixed-point iteration (AFPI) based on the semigroup property is developed for computing four extremal solutions of the discrete-time algebraic Riccati equation, which has not appeared in the existing literature. In addition, we prove that the convergence of the AFPI is at least R-suplinear with order r>1 under some mild assumptions. Numerical examples are shown to illustrate the feasibility and accuracy of the proposed algorithm.