For a generalized Brinkman-Forchheimer's equation under divergence-free and mixed boundary conditions, the stationary equilibrium problem and the inverse problem of shape optimal control are considered.For a convex, geometry-dependent objective function, the equilibrium-constrained optimization is treated with the help of an adjoint state within the Lagrange approach.The shape differentiability of a Lagrangian with respect to linearized shape perturbations is derived in the analytic form by the velocity method.A Hadamard representation of the shape derivative using boundary integrals is derived.Its applications to path-independent integrals and to the gradient descent method are illustrated.