In this talk, we analyze the Schwarz alternating method for unconstrained elliptic optimal control problems.  One important feature of the method in this case is that the local error propagation operators of the algorithm are not always nonexpansive operators under the energy norm, which is different from that of the elliptic equation case. We discuss the uniform convergence properties of the method in the continuous case first and then apply the arguments to the finite difference discretization case. In both cases, we prove that the Schwarz alternating method is convergent if its counterpart for an elliptic equation is convergent. Meanwhile, the convergence factor of the method for the elliptic equation under the maximum norm also gives a uniform upper bound (with respect to the regularization parameter $\alpha$) of the convergence factor of the method for the optimal control problem under the maximum norm. The extension to the one-level multiplicative Schwarz method and the one-level parallel additive Schwarz method is also given. Numerical results corroborate the theoretical results.