Wasserstein分布鲁棒线性二次控制中的均值-协方差 turnpike 性质
Mean-covariance turnpikes in Wasserstein distributionally robust linear-quadratic control
AI总结：
针对带经验扰动数据的离散时间随机线性系统的长时域Wasserstein分布鲁棒线性二次控制问题，通过构造混合策略证明了相关量的指数turnpike性质，可降低长时域鲁棒控制的计算成本。
AI中文摘要：
我们针对带有经验扰动数据的离散时间随机线性系统，研究长时域Wasserstein惩罚极小极大控制问题，其中对抗性扰动分布会诱导时变均值与协方差动态，使得标准turnpike论证无法直接应用。针对可能非中心化的数据，我们通过约化的凸-凹哈密顿鞍问题刻画一般非零的均值参考量。我们证明了均值状态、伴随变量、控制量、最坏情况扰动均值以及闭环协方差的时域一致双边指数turnpike估计，表明当时域较长时，这些量大部分时间会接近静态参考值。我们进一步构造了一种混合策略，将与时域无关的仿射反馈与终端层的有限时域引导相结合，证明其最坏情况成本缺口随终端层长度指数衰减，且在整个时域上一致，这有助于降低长时域鲁棒控制的计算成本。数值例子验证了这些估计及其对Wasserstein惩罚的依赖性。
英文摘要：
We study long-horizon Wasserstein-penalized minimax control for discrete-time stochastic linear systems with empirical disturbance data, in which adversarial disturbance distributions induce time-varying mean and covariance dynamics, making standard turnpike arguments not directly applicable. For possibly uncentered data, we characterize the generally nonzero mean reference through a reduced convex-concave Hamiltonian saddle problem. We prove horizon-uniform, two-sided exponential turnpike estimates for the mean state, adjoint, control, worst-case disturbance mean, and closed-loop covariance, showing that they spend the majority of time near static references when the horizon is long. We further construct a hybrid policy combining time-independent affine feedback with finite-horizon steering over a terminal layer, proving that its worst-case cost gap decays exponentially with the terminal-layer length uniformly in the horizon, which helps reducing the computation cost for long-horizon robust controls. Numerical examples illustrate the estimates and their dependence on the Wasserstein penalty.
