This paper considers the linear quadratic deterministic mean field control problem for large populations of heterogeneous agents, subject to convex state and input constraints, and coupled via a quadratic cost function which depends on the average population state. To control the optimal responses of the rational agents to a Nash equilibrium, we propose feedback iterative solutions based on operator theory arguments. Contrary to the state of the art, global convergence is ensured, under mild sufficient conditions on the matrices defining the cost functions, and not on the convex constraints.