Feedback Stabilization of the Three-Dimensional Navier-Stokes Equations using Generalized Lyapunov Equations

TL;DR

This paper introduces a novel approach using generalized Lyapunov equations to approximate the value function for feedback stabilization of 3D Navier-Stokes equations, overcoming non-differentiability issues.

Contribution

It proposes a new class of generalized Lyapunov equations with proven existence of unique solutions, enabling high-order approximation of the value function, states, and controls.

Findings

Existence of unique solutions to the new Lyapunov equations.

Feedback operators approximate the value function and controls to arbitrary order.

Method addresses non-differentiability of the value function in 3D Navier-Stokes stabilization.

Abstract

The approximation of the value function associated to a stabilization problem formulated as optimal control problem for the Navier-Stokes equations in dimension three by means of solutions to generalized Lyapunov equations is proposed and analyzed. The specificity, that the value function is not differentiable on the state space must be overcome. For this purpose a new class of generalized Lyapunov equations is introduced. Existence of unique solutions to these equations is demonstrated. They provide the basis for feedback operators, which approximate the value function, the optimal states and controls, up to arbitrary order.