Convergence Analysis for Splitting of the Abstract Riccati Equation

We consider a splitting-based approximation of the abstract Riccati equation in the setting of Hilbert–Schmidt operators. The Riccati equation arises in many different areas and is important within the field of optimal control. While convergence of different methods for approximating the Riccati equation is discussed in several studies, none of them rigorously prove an order of convergence. In this paper we conduct such a convergence analysis and show that the splitting method converges with the same order as the implicit Euler scheme. As it requires no Newton iterations we expect it to be more efficient than common non-splitting methods. We also show that the splitting method preserves low-rank structure in the matrix-valued case, which is essential for large-scale problems. A numerical experiment demonstrates the validity of our theory.