为随机控制问题提出新最优性条件，可加速求解。
Rough Stochastic Pontryagin Maximum Principle and an Indirect Shooting Method
- 基于粗糙微分方程推导出无需前后向SDE的最优性条件。
- 在稳定化任务中，新方法收敛速度比直接法快10倍。
- 适合需要高效求解随机控制问题的研究者使用。
我们为由高斯粗糙路径驱动的粗糙微分方程（RDE）系统，推导出确定性控制下的随机最优控制一阶庞特里亚金最优性条件。该庞特里亚金最大原理（PMP）适用于由布朗运动驱动的随机微分方程（SDE）系统，但不依赖前后向SDE，且使用与确定性情形相同的哈密顿量。证明通过利用高斯粗糙路径的最新成果，建立非线性和线性RDE解的可积误差界，再结合标准的针状变分技术完成。作为应用，我们提出了首个针对非线性随机最优控制的间接打靶法，并在稳定化任务中验证其收敛速度比直接法快10倍。
原文摘要 · Abstract (English)
We derive first-order Pontryagin optimality conditions for stochastic optimal control with deterministic controls for systems modeled by rough differential equations (RDE) driven by Gaussian rough paths. This Pontryagin Maximum Principle (PMP) applies to systems following stochastic differential equations (SDE) driven by Brownian motion, yet it does not rely on forward-backward SDEs and involves the same Hamiltonian as the deterministic PMP. The proof consists of first deriving various integrable error bounds for solutions to nonlinear and linear RDEs by leveraging recent results on Gaussian rough paths. The PMP then follows using standard techniques based on needle-like variations. As an application, we propose the first indirect shooting method for nonlinear stochastic optimal control and show that it converges 10x faster than a direct method on a stabilization task.
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