On the Numerical Solution of Large-scale Differential Riccati Equations

This thesis investigates efficient numerical methods for solving large-scale Differential Riccati Equations (DREs), which commonly arise in optimal control and filtering problems. The DRE is discretized in time using Backward Differentiation Formula (BDF) schemes, leading to a sequence of Algebraic Riccati Equations (AREs) solved via iterative techniques such as the Newton–Kleinman method, Newton’s method with exact line search, and low-rank ADI-based algorithms. A modified ADI scheme is then proposed, allowing for nonzero initial iterates and enabling the reuse of previous solutions to accelerate convergence. The performance of the low-rank methods is validated on the rail profile cooling problem, modeled within the Linear Quadratic Regulator (LQR) framework. Numerical experiments confirm the expected convergence order O(h^p) and demonstrate that the proposed low-rank approaches substantially improve computational efficiency and scalability for high-dimensional problems.