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Optimal boundary control for steady motions of a self-propelled body in a Navier-Stokes liquid

Consider a rigid body ${\mathcal S} \subset {\mathbb R}^3$ immersed in an infinitely extended Navier-Stokes liquid and the motion of the body-fluid interaction system described from a reference frame attached to ${\mathcal S}$. We are interested in steady motions of this coupled system, where the region occupied by the fluid is the exterior domain $Ω= {\mathbb R}^3 \setminus {\mathcal S}$. This paper deals with the problem of using boundary controls $v_*$, acting on the whole $\partialΩ$ or just on a portion $Γ$ of $\partialΩ$, to generate a self-propelled motion of ${\mathcal S}$ with a target velocity $V(x):=ξ+ω\times x$ and to minimize the drag about ${\mathcal S}$. 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 $v_*\subset Γ$, and for tangential controls, i.e, $v_*\cdot n|_{\partial Ω}=0$, where $n$ is the outward unit normal to $\partial Ω$. 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.