The State-Dependent Riccati Equation in Nonlinear Optimal Control: Analysis, Error Estimation and Numerical Approximation

TL;DR

This paper analyzes the state-dependent Riccati equation (SDRE) framework for nonlinear optimal control, clarifying its connection to the Hamilton-Jacobi-Bellman (HJB) equation and deriving residual-based error Bounds. It introduces an optimal semilinear decomposition strategy to minimize the SDRE residual and provides quantitative comparisons between SDRE and true HJB solutions. The work contrasts two numerical strategies—offline-online and Newton–Kleinman (including cascaded and hybrid variants)—and validates them through nonlinear reaction-diffusion PDE experiments, highlighting stability and computational trade-offs. The findings advocate NK-based approaches, especially cascaded or hybrid variants, for robust, real-time SDRE control and point to future directions in scalable solvers, low-rank techniques, and stochastic SDRE extensions.

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, demonstrating the superiority of the Newton-Kleinman approach in achieving stable and cost-effective solutions.

Figures (5)

• Figure 1: Van Der Pol example. Left: The temporal evolution of both the error between the value function and its SDRE approximation and the corresponding error bound as given in \ref{['eq:bound']} along the trajectory induced by the SDRE controller. Right: controlled solution using the SDRE controller.
• Figure 2: Allen-Cahn example. Variation of the residual as a function of the parameter $\alpha$ in the semilinear form, demonstrating the existence of at least one zero for $y_0(x) = \cos (\pi x)$ (left) and $y_0(x) = \sin (\pi x)$ (right).
• Figure 3: Zeldovich example. Left: Decay of the singular values of the Riccati solution $P(y_0)$ for the two different cases. Right: Off diagonal decay for the solution $P(y_0)$ in the second case.
• Figure 4: Zeldovich example (Case 1). Running cost for the controlled trajectories with the different techniques with $\mu =1$ (left) and $\mu=2$ (right).
• Figure 5: Zeldovich example (Case 2). Running cost for the controlled trajectories with the different techniques with $\mu =1$ (left) and $\mu=2$ (right).

Theorems & Definitions (17)

• Definition 2.1
• Definition 2.2
• Theorem 2.1
• Theorem 2.2
• proof
• Proposition 2.1
• proof
• Example 2.1: Linear Quadratic Regulator
• Example 2.2: Van Der Pol oscillator
• Proposition 3.1