This paper investigates the local convergence of the Lagrange-SQP-Newton method applied to an optimal control problem governed by a phase field equation with distributed control. The phase field equation is a system of two semilinear parabolic differential equations. Stability analysis of optimization problems and regularity results for parabolic differential equations are used to prove convergence of the controls with respect to the L²(Q) norm and with respect to the L^{infinity}(Q) norm.