Open loop stabilization of incompressible Navier–Stokes equations in a 2d channel using power series expansion

Abstract

In this article, we discuss the stabilization of incompressible Navier-Stokes equations in a 2d channel around a fluid at rest when the control acts only on the normal component of the upper boundary. In this case, the linearized equations are not controllable nor stabilizable at an exponential rate higher than νπ 2 /L 2 , when the channel is of width L and of length 2π and ν denotes the viscosity parameter. Our main result allows to go above this threshold and reach any exponential decay rate by using the non-linear term to control the directions which are not controllable for the linearized equations. Our approach therefore relies on writing the controlled trajectory as an expansion of order two taking the form εα + ε 2 β for some ε > 0 small enough. This method is inspired by the previous work [18] by J.-M. Coron and E. Crépeau on the controllability of the Korteweg de Vries equations.In this section, we recall some basic facts on the linearized equation (1.4) and give a modal description adapted to our setting.

Null controllability of the incompressible Stokes equations in a 2-D channel using normal boundary control

“…This also shows that ξ ′′′ k,l (1) = 0 for all k ∈ 2π L Z − {0}, and l ∈ N (see also Lemma 4.1 in [12] and Proposition 2.1 in [6]). Thus there exists a positive M k independent of l, such that |ξ ′′′ k,l (1)| ≥ M k k 2 e |k| |λ k,l ||µ k,l |, ∀ l ∈ N, ∀ |k| ≤ k, k = 0. where (φ k , ξ k , q k ) is the solution of (3.6) with terminal condition (φ k (·, T ), ξ k (·, T )) ∈ (L 2 (0, 1)) 2 .…”

“…In [6], Chowdhury and Ervedoza proved a local stabilization result for the viscous incompressible Navier-Stokes equations (1.1) at any exponential decay rate by a normal boundary control acting at the upper boundary. The linearized system around zero is exponentially stable with decay rate νπ 2 but not stabilizable at a higher decay rate.…”

Insensitizing controls for the heat equation with respect to boundary variations

“…Therefore, we use techniques specifically designed to deal with such kind of non-linearities, which consists in choosing the control functions in a vector space of much larger dimension than the number of constraints. Similarly to what has been done in another context for the stabilizability of the Navier-Stokes equation, see [10], if there are N constraints imposed by the exact insensitization for E, we look for control functions in a vector space of size (at most) 2N which is suitably designed. In particular, even if there is one constraint (i.e.…”

Quadratic behaviors of the 1D linear Schr{ö}dinger equation with bilinear control

“…When the linearized system is not controllable, the strategy of performing a power series expansion of the solution, presented in [18,Chap. 8] for finite-dimensional control system, can be used to prove positive controllability results as in [3,8,13,14,19] or stabilization results as in [16,21,22].…”