We study a few constrained Stochastic Optimal Control Problems. First, we look at problems with terminal constraints. For various convex problems with constrained control, such as Linear Quadratic Mean-Field problem or Non-Markovian problem with stochastic coefficients, we draw equivalence relationship between the Fritz John condition and Karush–Kuhn–Tucker (KKT) conditions. Then we construct an unconstrained problem with the Lagrange Multiplier derived from Fritz John condition. Finally, we show the equivalence between the optimality of the unconstrained problem and its original problem. Furthermore, we look at the Duality of Linear Quadratic Mean-Field control problems and find an equivalence relationship between the primal and dual problems in the absence of control constraints. Lastly we compare the Riccati solutions to the Linear Quadratic Mean-Field control problem and the empirical solutions to the Mean-Field Forward Backward Stochastic Differential Equations (FBSDEs) using Deep Learning to verify our results.