In this work, we study the optimal control of stochastic Burgers equation perturbed by Gaussian and Lévy-type noises with distributed control process acting on the state equation. We use dynamic programming approach for the feedback synthesis to obtain an infinite-dimensional second-order Hamilton–Jacobi–Bellman (HJB) equation consisting of an integro-differential operator with Lévy measure associated with the stochastic control problem. Using the regularizing properties of the transition semigroup corresponding to the stochastic Burgers equation and compactness arguments, we solve the HJB equation and the resultant feedback control problem.

在这项工作中，我们研究了受高斯和 Lévy 型噪声扰动的随机 Burgers 方程的最优控制，分布式控制过程作用于状态方程。我们使用动态规划方法进行反馈综合，以获得无限维二阶 Hamilton-Jacobi-Bellman (HJB) 方程，该方程由积分微分算子和与随机控制问题相关的 Lévy 测度组成。利用对应于随机 Burgers 方程和紧性参数的过渡半群的正则化性质，我们求解 HJB 方程和由此产生的反馈控制问题。