Closed-form SDRE control solution for second-order non-linear systems

Optimal control theory is widely used to develop control strategies that minimize a given cost function while ensuring both system stability and performance. Within this framework, Riccati Equation-based optimal control has emerged as an approach gaining increasing attention due to its remarkable effectiveness. In contrast to the linear case, determining analytical solutions of the Riccati equation for nonlinear systems remains a significant challenge. The common approach involves pointwise evaluation of system matrices and the repeated solution of the Riccati equation at each iteration step, a process that is computationally intensive and prone to numerical inaccuracies. We propose a mathematical approach that derives a symbolic closed-form solution to the State-Dependent Riccati Equation (SDRE) for second-order input-affine nonlinear systems, avoiding iterative computation. The method yields up to eight candidate analytical solutions, analysed based on structural and feasibility conditions. Moreover, necessary conditions to select the feasible solutions in generalized context are provided. Simulation results show that the solutions match or outperform the performance of established method with significantly reduced computational effort.