Control of underwater vehicles in inviscid fluids II. Flows with vorticity
Lecaros, Rodrigo1 ; Rosier, Lionel2
1 CMM – Centro de Modelamiento Matemático., Universidad de Chile (UMI CNRS 2807), Avenida Blanco Encalada 2120, Casilla 170-3, Correo 3, Santiago, Chile.
2 Centre Automatique et Systèmes, MINES ParisTech, PSL Research University, 60 Boulevard Saint-Michel, 75272 Paris cedex 06, France.
ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 1325-1352.

In a recent paper, the authors investigated the controllability of an underwater vehicle immersed in an infinite volume of an inviscid fluid, assuming that the flow was irrotational. The aim of the present paper is to pursue this study by considering the more general case of a flow with vorticity. It is shown here that the local controllability of the position and the velocity of the underwater vehicle (a vector in R 12 ) holds in a flow with vorticity whenever it holds in a flow without vorticity.

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2016040
Classification : 35Q35, 76B03, 93B05
Mots-clés : Underwater vehicle, fluid-structure interaction, Euler equations, vorticity, exterior domain, exact controllability, return method
Lecaros, Rodrigo 1 ; Rosier, Lionel 2

1 CMM – Centro de Modelamiento Matemático., Universidad de Chile (UMI CNRS 2807), Avenida Blanco Encalada 2120, Casilla 170-3, Correo 3, Santiago, Chile.
2 Centre Automatique et Systèmes, MINES ParisTech, PSL Research University, 60 Boulevard Saint-Michel, 75272 Paris cedex 06, France. 
@article{COCV_2016__22_4_1325_0,
     author = {Lecaros, Rodrigo and Rosier, Lionel},
     title = {Control of underwater vehicles in inviscid fluids {II.} {Flows} with vorticity},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1325--1352},
     publisher = {EDP-Sciences},
     volume = {22},
     number = {4},
     year = {2016},
     doi = {10.1051/cocv/2016040},
     zbl = {1354.35079},
     mrnumber = {3570504},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2016040/}
}
TY  - JOUR
AU  - Lecaros, Rodrigo
AU  - Rosier, Lionel
TI  - Control of underwater vehicles in inviscid fluids II. Flows with vorticity
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2016
SP  - 1325
EP  - 1352
VL  - 22
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2016040/
DO  - 10.1051/cocv/2016040
LA  - en
ID  - COCV_2016__22_4_1325_0
ER  - 
%0 Journal Article
%A Lecaros, Rodrigo
%A Rosier, Lionel
%T Control of underwater vehicles in inviscid fluids II. Flows with vorticity
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2016
%P 1325-1352
%V 22
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2016040/
%R 10.1051/cocv/2016040
%G en
%F COCV_2016__22_4_1325_0
Lecaros, Rodrigo; Rosier, Lionel. Control of underwater vehicles in inviscid fluids II. Flows with vorticity. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 1325-1352. doi : 10.1051/cocv/2016040. http://www.numdam.org/articles/10.1051/cocv/2016040/

J.-M. Coron, On the controllability of 2-D incompressible perfect fluids. J. Math. Pures Appl. 75 (1996) 155–188. | MR | Zbl

J.-M. Coron, Control and nonlinearity, vol. 136 of Math. Surveys Monographs. American Mathematical Society, Providence, RI (2007). | MR | Zbl

C. Conca, P. Cumsille, J. Ortega and L. Rosier, On the detection of a moving obstacle in an ideal fluid by a boundary measurement. Inverse Probl. 24 (2008) 045001. | DOI | MR | Zbl

C. Conca, M. Malik and A. Munnier, Detection of a moving rigid body in a perfect fluid. Inverse Probl. 26 (2010) 095010. | DOI | MR | Zbl

O. Glass and L. Rosier, On the control of the motion of a boat. Math. Models Methods Appl. Sci. 23 (2013) 617–670. | DOI | MR | Zbl

O. Glass, F. Sueur and T. Takahashi, Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid. Ann. Sci. Éc. Norm. Supér. 45 (2012) 1–51. | DOI | Numdam | MR | Zbl

V.I. Judovič, A two-dimensional non-stationary problem on the flow of an ideal incompressible fluid through a given region. Mat. Sb. (N.S.) 64 (1964) 562–588. | MR

T. Kato, M. Mitrea, G. Ponce and M. Taylor, Extension and representation of divergence-free vector fields on bounded domains. Math. Res. Lett. 7 (2000) 643–650. | DOI | MR | Zbl

A.V. Kazhikhov, Note on the formulation of the problem of flow through a bounded region using equations of perfect fluid. Prikl. Matem. Mekhan. 44 (1980) 947–950. | MR | Zbl

K. Kikuchi, The existence and uniqueness of nonstationary ideal incompressible flow in exterior domains in 𝐑 3 . J. Math. Soc. Japan 38 (1986) 575–598. | DOI | Zbl

R. Lecaros and L. Rosier, Control of underwater vehicles in inviscid fluids – I. Irrotational flows. ESAIM: COCV 20 (2014) 662–703. | Numdam | Zbl

J.H. Ortega, L. Rosier and T. Takahashi, Classical solutions for the equations modelling the motion of a ball in a bidimensional incompressible perfect fluid. ESAIM: M2AN 39 (2005) 79–108. | DOI | Numdam | Zbl

J.H. Ortega, L. Rosier and T. Takahashi, On the motion of a rigid body immersed in a bidimensional incompressible perfect fluid. Ann. Inst. Henri Poincaré, Anal. Non Lin. 24 (2007) 139–165. | DOI | Zbl

C. Rosier and L. Rosier, Smooth solutions for the motion of a ball in an incompressible perfect fluid. J. Funct. Anal. 256 (2009) 1618–1641. | DOI | Zbl

E.M. Stein, Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J. (1970). | Zbl

F. Sueur, A Kato type theorem for the inviscid limit of the Navier-Stokes equations with a moving rigid body. Commun. Math. Phys. 316 (2012) 783–808. | DOI | Zbl

Y. Wang and A. Zang, Smooth solutions for motion of a rigid body of general form in an incompressible perfect fluid. J. Differ. Eq. 252 (2012) 4259–4288. | DOI | Zbl

Cité par Sources :