Structure-Preserving Spectral Dynamic Programming on Compact Lie Groups

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 Fej\'er-type filter on $G$, finite-rank, positivity preserving and non-expansive, together with a convergence theorem at the rate $O(\sqrt\delta+\sqrt{\epsilon+1/(\delta N^2)})$ for Lipschitz data, where $\delta$ is the time step, $N$ the spectral resolution and $\epsilon$ the viscosity. The viscosity may be zero, and the coupling $\delta=N^{-1}$ 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.