  We present new Dirichlet-Neumann and Neumann-Dirichlet algorithms with a time
domain decomposition applied to unconstrained parabolic optimal control
problems. After a spatial semi-discretization, we use the Lagrange multiplier
approach to derive a coupled forward-backward optimality system, which can then
be solved using a time domain decomposition. Due to the forward-backward
structure of the optimality system, three variants can be found for the
Dirichlet-Neumann and Neumann-Dirichlet algorithms. We analyze their
convergence behavior and determine the optimal relaxation parameter for each
algorithm. Our analysis reveals that the most natural algorithms are actually
only good smoothers, and there are better choices which lead to efficient
solvers. We illustrate our analysis with numerical experiments.
