This study introduces an approach to obtain a neighboring extremal optimal control (NEOC) solution for a closed-loop optimal control problem, applicable to a wide array of nonlinear systems and not necessarily quadratic performance indices. The approach involves investigating the variation incurred in the functional form of a known closed-loop optimal control law due to small, known parameter variations in the system equations or the performance index. The NEOC solution can formally be obtained by solving a linear partial differential equation, akin to those encountered in the iterative solution of a nonlinear Hamilton-Jacobi equation. Motivated by numerical procedures for solving these latter equations, we also propose a numerical algorithm based on the Galerkin algorithm, leveraging the use of basis functions to solve the underlying Hamilton-Jacobi equation of the original optimal control problem. The proposed approach simplifies the NEOC problem by reducing it to the solution of a simple set of linear equations, thereby eliminating the need for a full re-solution of the adjusted optimal control problem. Furthermore, the variation to the optimal performance index can be obtained as a function of both the system state and small changes in parameters, allowing the determination of the adjustment to an optimal control law given a small adjustment of parameters in the system or the performance index. Moreover, in order to handle large known parameter perturbations, we propose a homotopic approach that breaks down the single calculation of NEOC into a finite set of multiple steps. Finally, the validity of the claims and theory is supported by theoretical analysis and numerical simulations.

翻译：本研究提出了一种获取闭环最优控制问题邻域极值最优控制（NEOC）解的方法，该方法适用于各类非线性系统，且不要求性能指标必须为二次型。该方法的思路是：分析已知的闭环最优控制律因其函数形式受系统方程或性能指标中微小已知参数变化的影响而产生的变分。NEOC解可通过求解一个线性偏微分方程正式获得——该方程类似于非线性哈密顿-雅可比方程迭代求解过程中所遇到的方程形式。受此类方程数值求解方法的启发，我们进一步提出了一种基于伽辽金算法的数值方法，该方法利用基函数求解原最优控制问题的哈密顿-雅可比方程。所提方法将NEOC问题简化为求解一组简单的线性方程，从而无需对调整后的最优控制问题进行完整重解。此外，最优性能指标的变分可表示为系统状态与参数微小变化的函数，从而允许在系统或性能指标参数发生微小调整时确定最优控制律的调整量。为处理较大的已知参数扰动，我们进一步提出了一种同伦方法，将单次NEOC计算分解为有限步骤的序列计算。最后，理论分析与数值仿真验证了所述理论与方法的有效性。