This work analyzes an optimal control problem for the Navier-Stokes-Brinkman equations, where the control is part of the state equations as a permeability coefficient, motivated by the need to find alternative techniques to shape and topology optimization to identify obstacles and domain deformations. After an analysis of first and second order optimality conditions and existence of an optimal control, a finite element method will be presented to obtain approximations of the optimal control and the states from a variational formulation, with a conformal approximation for the state equation, with its respective a priori error analysis. We also present a residual a posteriori error estimator that is efficient and reliable. The theoretical findings are complemented with numerical examples that verify the error estimates. Finally, an experiment is presented that seeks to identify an obstacle immersed in a fluid from partial measurements of the fluid velocity using an adaptive refinement strategy.