In this article, we are concerned about the velocity tracking optimal control problem for 3D critical convective Brinkman–Forchheimer equations defined on a simply connected bounded domain \(\mathfrak {D} \subset \mathbb {R}^3\) with \(\textrm{C}^2\) -boundary \(\partial \mathfrak {D}\) . The control is introduced through an external force. The objective is to optimally minimize a velocity tracking cost functional, for which the velocity vector field is oriented towards a target velocity. Most importantly, we are concerned about the first-order necessary optimality conditions for above-mentioned optimal control problem which is the main challenging task of this article. To overcome the difficulties related to the differentiability of the control-to-state mapping, consequence of the lack of regularity of the state variable on bounded domains, we first establish some intermediate optimality conditions and then pass to the limit.
