Approximate Lagrangian controllability for the 2-D Euler equation. Application to the control of the shape of vortex patches

Abstract

In this paper, we consider the two-dimensional Euler equation in a bounded domain Ω, with a boundary control located on an arbitrary part of the boundary. We prove that, given two Jordan curves which are homotopic in Ω and which surround the same area, given an initial data and a positive time T , one can find a control such that the corresponding solution drives the first curve inside Ω arbitrarily close to the second one (in any C k norm) at time T. We also prove that given two vortex patches satisfying the same conditions on their contour, one can approximately deform the first one into the second one. Résumé. Dans cet article, nous considérons l'équation d'Euler des fluides parfaits incompressibles dans un domaine borné bidimensionnel, avec un contrôle frontière localisé sur une partie arbitraire du bord. Nous montrons qu'étant donnés deux courbes de Jordan homotopes et encerclant la même aire, une donnée initiale et un temps T strictement positif, on peut trouver un contrôle tel que la solution correspondante de l'équation d'Euler mène la première courbe vers la seconde de manière arbitrairement proche (en toute norme C k) au temps T. Nous montronségalement que sous la même condition sur les contours, on peut déformer de manière approchée une poche de tourbillon sur une autre dans le domaine.

A result concerning the global approximate controllability of the Navier–Stokes system in dimension 3

“…An important point in the study of the controllability of Navier-Stokes systems has been the so-called return method, introduced by J.-M. Coron in [2] to study the stabilization of some control systems and then used in [3] to prove global exact controllability for the Euler equations in dimension 2 (see also [12]) and then in [4] to prove global approximate controllability result for the Navier-Stokes system with Navier-slip boundary conditions (or when the control is acting on the whole boundary; this result was generalized later in [5] for the case of the Navier-Stokes system on a manifold without boundary). For the analysis of the similar three-dimensional situation for the Euler equations, see [10].…”

Some inverse and control problems for fluids

“…What has been proven in [32] is the following: This result is sharp in the sense that some counter-examples can be exhibited if one does not relax the condition (3.8).…”

“…Glass and T. Horsin [32] have extended the notion of controllability to the Lagrangian description of two-dimensional perfect fluids; the case of three-dimensionnal perfect fluids has also been studied in [33]. Previously, some toy models were investigated in [39] and [40].…”

“…This enabled O. Glass and T. Horsin in [32] to give counter-examples to the exact Lagrangian controllability that do not respect the condition (3.7). In fact, one can omit or relax (3.7), but this may imply that the controlled part of the fluid is allowed to leave Ω which, concerning possible applications to moving pollutants, would not be admissible.…”

“…To this purpose, the following is established in [32] and [33]: Theorem 3.6. Under the hypotheses made on γ 0 and γ 1 , there exists…”