Dynamic Programming of Stochastic 2-D Navier-Stokes Equations Forced by Levy Noise

TL;DR

This paper develops a dynamic programming framework for optimal feedback control of stochastic 2D Navier-Stokes equations influenced by Levy noise, addressing the associated complex Hamilton-Jacobi-Bellman equation.

Contribution

It introduces a novel approach to solve the infinite-dimensional HJB equation with Levy noise, providing smooth solutions for stochastic fluid control problems.

Findings

Existence of smooth solutions to the HJB equation in weighted spaces

Effective feedback control synthesis for stochastic Navier-Stokes equations

Application of transition semigroup regularization properties

Abstract

In this article, we study optimal feedback control synthesis of stochastic 2D Navier-Stokes equations perturbed Levy type noise with distributed stochastic control process acting on the state equation. We use the dynamic programming approach to solve this control problem which involves the study of second order infinite dimensional Hamilton- Jacobi-Bellman (HJB) equation consisting of an integro-differential operator with Levy measure associated with the stochastic control problem. Using the regularizing properties of the transition semigroup corresponding to the stochastic 2D Navier-Stokes equation, we obtain a smooth solution in weighted function space for the HJB equation and solve the resultant feedback control problem.