标题： 一种最优控制面的多维傅里叶逼近 显示英文标题

标题： A Multidimensional Fourier Approximation of Optimal Control Surfaces

摘要： 这项工作考虑了使用多变量傅里叶级数近似初始条件和时变最优控制及轨迹表面的问题。提出了一种改进的增广拉格朗日算法，将最优控制问题转化为无约束优化问题，并解决了两个问题：牛顿力学背景下的二次控制问题，以及由复制者动力学支配的奇循环博弈中出现的控制问题。给出了各种计算结果。探讨了自动微分的使用，以绕过一阶优化过程中的复杂梯度计算。此外，推导了在一维和二维傅里叶级数近似情况下的均方误差界，从而得出$n$维问题的一般界。 显示英文摘要

摘要： This work considers the problem of approximating initial condition and time-dependent optimal control and trajectory surfaces using multivariable Fourier series. A modified Augmented Lagrangian algorithm for translating the optimal control problem into an unconstrained optimization one is proposed and two problems are solved: a quadratic control problem in the context of Newtonian mechanics, and a control problem arising from an odd-circulant game ruled by the replicator dynamics. Various computational results are presented. Use of automatic differentiation is explored to circumvent the elaborated gradient computation in the first-order optimization procedure. Furthermore, mean square error bounds are derived for the case of one and two-dimensional Fourier series approximations, inducing a general bound for problems of $n$ dimensions.