In this paper, we improve an earlier iterative successive approximation method for solving the Hamilton–Jacobi equation (HJE) arising in deterministic optimal control of affine nonlinear systems. The new methods generate smooth approximate solutions for systems with polynomial nonlinearities, compared with the former method that generates rational functions with possible singularities in the domain. We prove quadratic convergence of the methods and demonstrate their effectiveness with some examples. Application to factorisation of nonlinear systems is also discussed.

中文翻译：

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仿射非线性系统最优控制问题中解哈密顿-雅可比方程的改进迭代计算方法

在本文中，我们改进了一种较早的迭代逐次逼近方法，用于求解在仿射非线性系统的确定性最优控制中出现的 Hamilton-Jacobi 方程 (HJE)。新方法为具有多项式非线性的系统生成平滑的近似解，而前一种方法生成域中可能具有奇点的有理函数。我们证明了这些方法的二次收敛性，并通过一些例子证明了它们的有效性。还讨论了非线性系统分解的应用。