Inheritance properties for projection methods on continuous-time algebraic Riccati equations

In this paper, block Krylov-based methods are considered to compute a low rank approximation to the unique stabilizing solution of large-scale continuous-time algebraic Riccati equations, arising from linear-quadratic regulator control problems for linear time-invariant systems and second-order systems, respectively. When the original system is stabilizable and detectable, two necessary and sufficient conditions in terms of some nullspaces of small-sized matrices are respectively derived for the stabilizability and detectability of projected matrix pairs, and thus analogous results can be extended to second-order models. Furthermore, we provide an algorithm to examine the inheritance properties of these matrix pairs based on the theoretical results and an equivalent technique of evaluation for block rational Arnoldi methods. Numerical examples are given to demonstrate the performance of the rational Krylov subspace method with one single shift and the feasibility of our proposed algorithm as well.