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：本文提出了一种称为Y算子的新型算子，旨在提升基于Actor-Critic(AC)框架的强化学习在随机微分方程(SDEs)控制系统中的控制性能。该算子巧妙地将一类母子系统的随机性融入Critic网络的损失函数，显著提升了强化学习算法的控制性能。此外，Y算子将状态值函数偏微分方程的求解问题，优雅地转化为系统SDE中漂移函数与扩散函数的并行求解问题，其有效性已通过严格的数学证明予以确认。这一转换使得基于Y算子的强化学习框架能够高效处理基于模型和数据驱动两类系统中的最优控制问题。通过线性和非线性数值算例，本文验证了所提方法在收敛后性能优于现有方法，展现了YORL框架的优越性。