In this paper, we aim to develop the theory of optimal stochastic control for branching diffusion processes where both the movement and the reproduction of the particles depend on the control. More precisely, we study the problem of minimizing a criterion that is expressed as the expected value of the product of individual costs penalizing the final position of each particle. In this setting, we show that the value function is the unique viscosity solution of a nonlinear parabolic PDE, that is, the Hamilton-Jacobi-Bellman equation corresponding to the problem. To this end, we extend the dynamic programming approach initiated by Nisio to deal with the lack of independence between the particles as well as between the reproduction and the movement of each particle. In particular, we exploit the particular form of the optimization criterion to recover a weak form of the branching property. In addition, we provide a precise formulation and a detailed justification of the adequate dynamic programming principle.