Dynamic Programming of the Stochastic Burgers Equation Driven by Lévy Noise

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.