This paper introduces a novel operator, termed the Y operator, to elevate control performance in Actor-Critic(AC) based reinforcement learning for systems governed by stochastic differential equations(SDEs). The Y operator ingeniously integrates the stochasticity of a class of child-mother system into the Critic network's loss function, yielding substantial advancements in the control performance of RL algorithms.Additionally, the Y operator elegantly reformulates the challenge of solving partial differential equations for the state-value function into a parallel problem for the drift and diffusion functions within the system's SDEs.A rigorous mathematical proof confirms the operator's validity.This transformation enables the Y Operator-based Reinforcement Learning(YORL) framework to efficiently tackle optimal control problems in both model-based and data-driven systems.The superiority of YORL is demonstrated through linear and nonlinear numerical examples showing its enhanced performance over existing methods post convergence.

翻译：本文针对随机微分方程(SDEs)驱动的Actor-Critic(AC)强化学习系统，提出了一种新型算子——Y算子，通过将一类母子系统的随机性巧妙融入Critic网络的损失函数，实现了控制性能的显著提升。该算子将状态价值函数偏微分方程的求解问题，优雅地转化为系统SDEs中漂移函数与扩散函数的并行求解问题，并通过严谨的数学证明验证了其有效性。基于此变换，Y算子强化学习(YORL)框架能够高效解决基于模型与数据驱动系统中的最优控制问题。线性与非线性数值算例表明，YORL在收敛后相较于现有方法展现出更优的控制性能。