This paper is concerned with dynamic domain decomposition for optimal boundary control and for approximate and exact boundary controllability of wave propagation in heterogeneous media. We consider a cost functional which penalizes the deviation of the final state of the solution of the global problem from a specified target state. For any fixed value of the penalty parameter, optimality conditions are derived for both the global optimal control problem and for local optimal control problems obtained by a domain decomposition and a saddle-point-type iteration. Convergence of the iterations to the solution of the global optimality system is established. We then pass to the limit in the iterations as the penalty parameter increases without bound and show that the limiting local iterations converge to the solution of the optimality system associated with the problem of finding the minimum norm control that drives the solution of the global problem to a specified target state.