Consider a rigid body 𝒮 ⊂ ℝ3 immersed in an infinitely extended Navier-Stokes liquid and the motion of the body-fluid interaction system described from a reference frame attached to 𝒮. We are interested in steady motions of this coupled system, where the region occupied by the fluid is the exterior domain Ω = ℝ3 \ 𝒮. This paper deals with the problem of using boundary controls v*, acting on the whole ∂Ω or just on a portion Γ of ∂Ω, to generate a self-propelled motion of 𝒮 with a target velocity V (x) := ξ + ω × x and to minimize the drag about 𝒮. Firstly, an appropriate drag functional is derived from the energy equation of the fluid and the problem is formulated as an optimal boundary control problem. Then the minimization problem is solved for localized controls, such that supp v*⊂ Γ, and for tangential controls, i.e, v*⋅ n|$$ = 0, where n is the outward unit normal to ∂Ω. We prove the existence of optimal solutions, justify the Gâteaux derivative of the control-to-state map, establish the well-posedness of the corresponding adjoint equations and, finally, derive the first order optimality conditions. The results are obtained under smallness restrictions on the objectives |ξ| and |ω| and on the boundary controls.
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DOI : 10.1051/cocv/2020073
Mots-clés : 3-D Navier-Stokes equations, exterior domain, rotating body, self-propelled motion, boundary control, drag reduction
@article{COCV_2020__26_1_A92_0,
author = {Hishida, Toshiaki and Silvestre, Ana Leonor and Takahashi, Tak\'eo},
title = {Optimal boundary control for steady motions of a self-propelled body in a {Navier-Stokes} liquid},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {26},
year = {2020},
doi = {10.1051/cocv/2020073},
mrnumber = {4175379},
zbl = {1459.76048},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2020073/}
}
TY - JOUR AU - Hishida, Toshiaki AU - Silvestre, Ana Leonor AU - Takahashi, Takéo TI - Optimal boundary control for steady motions of a self-propelled body in a Navier-Stokes liquid JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2020073/ DO - 10.1051/cocv/2020073 LA - en ID - COCV_2020__26_1_A92_0 ER -
%0 Journal Article %A Hishida, Toshiaki %A Silvestre, Ana Leonor %A Takahashi, Takéo %T Optimal boundary control for steady motions of a self-propelled body in a Navier-Stokes liquid %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2020073/ %R 10.1051/cocv/2020073 %G en %F COCV_2020__26_1_A92_0
Hishida, Toshiaki; Silvestre, Ana Leonor; Takahashi, Takéo. Optimal boundary control for steady motions of a self-propelled body in a Navier-Stokes liquid. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 92. doi : 10.1051/cocv/2020073. http://www.numdam.org/articles/10.1051/cocv/2020073/
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