[Paper Review] Feedback Stabilization of the Two-Dimensional Navier-Stokes Equations by Value Function Approximation

Tobias Breiten, Karl Kunisch+1 more|arXiv (Cornell University)|Feb 1, 2019

Stability and Controllability of Differential Equations49 references20 citations

TL;DR

This paper establishes the smoothness of the value function for infinite-horizon optimal control of the 2D Navier-Stokes equations around a steady state, derives its Taylor expansion to construct polynomial feedback laws, and proves convergence rates for the resulting stabilizing controls and closed-loop systems. The value function's derivatives satisfy a Riccati equation (order 2) and generalized Lyapunov equations (higher orders), enabling efficient feedback stabilization via approximation.

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.

Motivation & Objective

• To analyze the regularity of the value function for infinite-horizon optimal control of the 2D Navier-Stokes equations.
• To characterize the value function as a solution to a Hamilton-Jacobi-Bellman (HJB) equation in the classical sense, not just in viscosity sense.
• To derive polynomial feedback laws through Taylor expansion of the value function around a steady state.
• To establish convergence rates for the resulting feedback controls and the corresponding closed-loop systems.
• To demonstrate that the second-order derivatives of the value function solve an algebraic Riccati equation, while higher-order derivatives satisfy generalized Lyapunov equations.

Proposed method

• The value function is shown to be smooth in a neighborhood of a steady state, enabling classical solution of the HJB equation.
• Second-order derivatives of the value function are proven to satisfy an algebraic Riccati equation.
• Higher-order derivatives are shown to satisfy generalized Lyapunov equations arising from the nonlinear structure of the Navier-Stokes equations.
• A feedback control law is constructed via Taylor expansion of the value function up to a given order.
• The closed-loop system's well-posedness is established using specialized functional analytic estimates tailored to the nonlinear terms in the Navier-Stokes equations.
• Convergence rates for the feedback control and the resulting closed-loop system are derived based on the approximation error of the value function.

Experimental results

Research questions

• RQ1 Is the value function for the infinite-horizon optimal control problem of the 2D Navier-Stokes equations smooth around a steady state?
• RQ2 Can the derivatives of the value function be characterized as solutions to Riccati and generalized Lyapunov equations?
• RQ3 Does a Taylor expansion of the value function yield a stabilizing feedback law with quantifiable convergence rates?
• RQ4 How do the nonlinear terms in the Navier-Stokes equations affect the structure of the generalized Lyapunov equations for higher-order derivatives?
• RQ5 What is the convergence rate of the feedback control and closed-loop system when approximating the value function via Taylor expansion?

Key findings

• The value function is smooth in a neighborhood of the steady state, and the HJB equation is satisfied in the classical sense, not just in the viscosity sense.
• The second-order derivatives of the value function solve an algebraic Riccati equation.
• Higher-order derivatives satisfy generalized Lyapunov equations specific to the nonlinear structure of the 2D Navier-Stokes equations.
• A polynomial feedback law is derived from the Taylor expansion of the value function, providing a constructive method for feedback stabilization.
• The resulting feedback control stabilizes the system, and a convergence rate is established for both the control and the closed-loop system.
• The well-posedness of the closed-loop system is guaranteed under small initial perturbations, with the control acting through a bounded linear operator on the state.