The State-Dependent Riccati Equation in Nonlinear Optimal Control: Analysis and Numerical Approximation
分析了状态相关Riccati方程方法的理论基础、误差界和数值逼近技术，通过非线性反应扩散PDE控制实验比较了两种求解方法的性能，对控制理论研究者有参考价值。
Abstract The State-Dependent Riccati Equation (SDRE) approach is extensively utilized in nonlinear optimal control as a reliable framework for designing robust feedback control strategies. This work provides an analysis of the SDRE approach, examining its theoretical foundations, error bounds, and numerical approximation techniques. We explore the relationship between SDRE and the Hamilton-Jacobi-Bellman (HJB) equation, deriving residual-based error estimates to quantify its suboptimality. Additionally, we introduce an optimal semilinear decomposition strategy to minimize the residual. From a computational perspective, we analyze two numerical methods for solving the SDRE: the offline–online approach and the Newton–Kleinman iterative method. Their performance is assessed through a numerical experiment involving the control of a nonlinear reaction-diffusion PDE. Results highlight the trade-offs between computational efficiency and accuracy, indicating better performance of the Newton–Kleinman approach in achieving stable and cost-effective solutions in the reported experiments.