# Feedback Stabilization of the Two-Dimensional Navier-Stokes Equations by Value Function Approximation

## TL;DR

This paper analyzes the value function for optimal control of 2D Navier-Stokes equations, establishing smoothness, deriving Riccati and Lyapunov equations for derivatives, and proposing an approximation method with proven convergence.

## Contribution

It introduces a novel approach to approximate optimal feedback control for 2D Navier-Stokes equations using Taylor expansion of the value function, with convergence guarantees.

## Key findings

- Value function is smooth around a steady state.
- Derivatives satisfy Riccati and Lyapunov equations.
- Approximate feedback law converges to the optimal control.

## Abstract

The value function associated with an optimal control problem subject to the Navier-Stokes equations in dimension two is analyzed. Its smoothness is established around a steady state, moreover, its derivatives are shown to satisfy a Riccati equation at the order two and generalized Lyapunov equations at the higher orders. An approximation of the optimal feedback law is then derived from the Taylor expansion of the value function. A convergence rate for the resulting controls and closed-loop systems is demonstrated.