AI 中文总结
研究连通紧致李群\(G\)上有限时域最优控制的谱逼近问题，通过用马尔可夫型谱滤波器代替正交投影恢复离散结构，给出费耶尔型滤波器及收敛速率，数值实验验证了结果并量化相关特性。
AI 中文摘要
我们研究了连通紧致李群\(G\)上有限时域最优控制的动态规划半群及其相关一阶哈密顿 - 雅可比 - 贝尔曼方程的谱逼近。贝尔曼算子在 supremum 范数下是单调且非扩张的，而\(L^{2}(G)\)的彼得 - 外尔分解是正交的，这导致了量化的不匹配。伽辽金迭代的自然 sup - 范数误差递归在每一步都会因谱投影的勒贝格常数而放大。我们通过用马尔可夫型谱滤波器代替正交投影来恢复离散层面的动态规划结构。一个辅助热核/消失粘性方案对于利普希茨数据产生定性的 sup - 范数收敛。主要结果是\(G\)上的一个费耶尔型滤波器，具有有限秩、保正性且非扩张，对于利普希茨数据有\(O(\sqrt\delta+\sqrt{\epsilon + 1/(\delta N^2)})\)的收敛速率，其中\(\delta\)是时间步长，\(N\)是谱分辨率，\(\epsilon\)是粘性。数值实验证实了预测的速率和滤波器偏差，并量化了两个基于图表的基线的框架依赖性。
英文摘要
We study spectral approximations of the dynamic programming semigroup for finite-horizon optimal control on a connected compact Lie group $G$, and of the associated first-order Hamilton-Jacobi-Bellman equation. The Bellman operator is monotone and non-expansive in the supremum norm, while the Peter-Weyl decomposition of $L^{2}(G)$, on which every Fourier method on $G$ rests, is orthogonal, and the mismatch is quantitative. The natural sup-norm error recursion of the Galerkin iteration is amplified at every step by the Lebesgue constant of the spectral projection, which grows logarithmically on $S^{1}$ and polynomially on compact Lie groups of rank one, including $\mathrm{SO}(3)$, and in computation the iteration violates elementary bounds within a few steps. We restore the dynamic programming structure at the discrete level by replacing the orthogonal projection with spectral filters of Markov type. An auxiliary heat-kernel/vanishing-viscosity scheme yields qualitative sup-norm convergence for Lipschitz data. The main result is a Fejér-type filter on $G$, finite-rank, positivity preserving and non-expansive, together with a convergence theorem at the rate $O(\sqrtδ+\sqrt{ε+1/(δN^2)})$ for Lipschitz data, where $δ$ is the time step, $N$ the spectral resolution and $ε$ the viscosity. The viscosity may be zero, and the coupling $δ=N^{-1}$ then gives the rate $N^{-1/2}$. The proof interprets the filter as a small random perturbation of the controlled dynamics, requires neither a priori regularity of the value function nor a consistency argument in the viscosity sense for the filtering step, and extends to a fully discrete realization based on positive cubature, with exact Wigner transport on $\mathrm{SO}(3)$. Numerical experiments confirm the predicted rates and filter bias and quantify the frame dependence of two chart-based baselines.
