  Optimization with time-dependent partial differential equations (PDEs) as
constraints {appears} in many science and engineering applications. The
associated first-order necessary optimality system consists of one forward and
one backward time-dependent PDE coupled with optimality conditions. An
optimization process by using the one-shot method determines the optimal
control, state and adjoint state at once, with the cost of solving a large
scale, fully discrete optimality system. Hence, such {a} one-shot method could
easily become computationally prohibitive when the time span is long or time
step is small. To overcome this difficulty, we propose several time domain
decomposition algorithms for improving the {computational efficiency of the
one-shot method}. In these algorithms, the optimality system is split into many
small subsystems over a much smaller time interval, which are coupled by
appropriate continuity matching conditions. Both one-level and two-level
multiplicative and additive Schwarz algorithms are developed for iteratively
solving the decomposed subsystems in parallel. In particular, the convergence
of the one-level, non-overlapping algorithms is proved. The effectiveness of
our proposed algorithms is demonstrated by both 1D and 2D numerical
experiments, where the developed two-level algorithms {show convergence rates
that are scalable} with respect to the number of subdomains.
