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In this paper we consider a fluid-structure interaction problem given by thesteady Navier Stokes equations coupled with linear elasticity taken from[Lasiecka, Szulc, and Zochoswki, Nonl. Anal.: Real World Appl., 44, 2018]. Anelastic body surrounded by a liquid in a rectangular domain is deformed by theflow which can be controlled by the Dirichlet boundary condition at the inlet.On the walls along the channel homogeneous Dirichlet boundary conditions and onthe outflow boundary do-nothing conditions are prescribed. We recall existenceresults for the nonlinear system from that reference and analyze the control tostate mapping generalizing the results of [Wollner and Wick, J. Math. FluidMech., 21, 2019] to the setting of the nonlinear Navier-Stokes equation for thefluid and the situation of mixed boundary conditions in a domain with corners.

Differentiability properties for boundary control of fluid-structure  interactions of linear elasticity with Navier-Stokes equations with  mixed-boundary conditions in a channel