SIAM Journal on Control and Optimization, Volume 59, Issue 1, Page 509-533, January 2021.
We consider the use of the polynomial chaos method to approximate optimal control problems with randomly varying uncertain parameters by deterministic surrogate problems. Our focus is on nonlinear problems that require a high expansion order to give meaningful statements about the uncertainty propagation. Their resulting size and complexity pose a computational challenge for traditional optimal control methods. Our contributions include an adaptive optimization strategy which refines the approximation quality separately for each state variable using suitable error estimates. The benefits are twofold: we obtain additional means for solution verification and reduce the computational effort for finding an approximate solution with increased precision, as is highlighted in a numerical case study with two nonlinear real-world problems. The algorithmic contribution is complemented by a convergence proof showing that the optimal control solutions approach the correct solution for increasing expansion orders.

中文翻译：

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不确定条件下直接最优控制的多项式混沌方法的收敛性分析和自适应阶数选择

SIAM控制与优化杂志，第59卷，第1期，第509-533页，2021年1月。
我们考虑使用多项式混沌方法通过确定性替代问题来近似估计具有随机变化的不确定参数的最优控制问题。我们的重点是非线性问题，这些问题需要高阶展开才能给出有关不确定性传播的有意义的陈述。它们的大小和复杂性给传统的最佳控制方法带来了计算上的挑战。我们的贡献包括自适应优化策略，该策略使用适当的误差估计分别为每个状态变量优化近似质量。好处是双重的：我们获得了用于解决方案验证的其他方法，并减少了以更高的精度查找近似解决方案的计算量，这在具有两个非线性现实问题的数值案例研究中得到了强调。