Optimal boundary control for steady motions of a self-propelled body in a Navier-Stokes liquid
Résumé
Consider a rigid body $S ⊂R3$ immersed in an infinitely extended Navier-Stokes liquid and the motion ofthe body-fluid interaction system described from a reference frame attached toS. We are interested in steadymotions 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 controlsv∗, acting on the whole∂Ω or just on a portionΓ of∂Ω, to generate a self-propelled motion ofSwith a target velocityV(x) :=ξ+ω×xand to minimizethe drag aboutS. Firstly, an appropriate drag functional is derived from the energy equation of the fluidand the problem is formulated as an optimal boundary control problem. Then the minimization problem issolved for localized controls, such that suppv∗⊂Γ, and for tangential controls, i.e,v∗·n|∂Ω= 0, wherenisthe outward unit normal to∂Ω. We prove the existence of optimal solutions, justify the Gˆateaux derivativeof 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 theobjectives|ξ|and|ω|and on the boundary controls
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