Internal Exponential Stabilization to a Nonstationary Solution for 3D Navier–Stokes Equations

Abstract

We consider the Navier-Stokes system in a bounded domain with a smooth boundary. Given a sufficiently regular time-dependent global solution, we construct a finite-dimensional feedback control that is supported by a given open set and stabilizes the linearized equation. The proof of this fact is based on a truncated observability inequality, the regularizing property for the linearized equation, and some standard techniques of the optimal control theory. We then show that the control constructed for the linear problem stabilizes locally also the full Navier-Stokes system.

Control and mixing for 2D Navier-Stokes equations with space-time localised noise

“…We now turn to the proof of Theorem 3.1. As was mentioned in the Introduction, a weaker version of this result was established in the paper [BRS11], and its proof repeats essentially the argument used there. However, since the finite-dimensionality in time for the control and the regularity and Lipschitz properties of Φ are important for the stochastic part of this work, we present a rather complete proof of Theorem 3.1.…”

“…In this section, we outline the proof of Theorem 2.1, which is based on two key ingredients: a coupling approach developed in [KS01, Mat02, KPS02, MY02, Hai02, Shi04] in the context of stochastic PDE's and a property of stabilisation to a non-stationary solution of Navier-Stokes equations [BRS11]. We first recall an abstract result established in [Shi08].…”

Optimal feedback control for undamped wave equations by solving a HJB equation

“…In [13] the authors consider an infinite horizon problem and propose a method using formally a quadratic Taylor approximation of the solution of the HJB equation. For a non-autonomous system that appears in the internal stabilization of Navier Stokes equation to a nonstationary solution a feedback law satisfying a differential Riccati equation can be found in [5].…”

Remarks on the Internal Exponential Stabilization to a Nonstationary Solution for 1D Burgers Equations

“…Let us be given a positive constant λ > 0, a continuous Lipschitz function χ ∈ W 1,∞ (Ω, R) with nonempty support, and a solutionû ∈ W st of (1) with ζ = 0, in a suitable Banach space W st . Then, following the procedure presented in [BRS11], we can prove that there exists an integer M = M (|û| W st ), a function η = η(t, x), defined for t > 0, x ∈ Ω, such that the solution u = u(t, x) of problem (1), with ζ = χP M η, and supplemented with the initial condition That is, the internal control ζ = χP M η stabilizes exponentially, with rate λ, the Burgers system to the reference trajectoryû.…”