  The ParaOpt algorithm was recently introduced as a time-parallel solver for
optimal-control problems with a terminal-cost objective, and convergence
results have been presented for the linear diffusive case with implicit-Euler
time integrators. We reformulate ParaOpt for tracking problems and provide
generalized convergence analyses for both objectives. We focus on linear
diffusive equations and prove convergence bounds that are generic in the time
integrators used. For large problem dimensions, ParaOpt's performance depends
crucially on having a good preconditioner to solve the arising linear systems.
For the case where ParaOpt's cheap, coarse-grained propagator is linear, we
introduce diagonalization-based preconditioners inspired by recent advances in
the ParaDiag family of methods. These preconditioners not only lead to a
weakly-scalable ParaOpt version, but are themselves invertible in parallel,
making maximal use of available concurrency. They have proven convergence
properties in the linear diffusive case that are generic in the time
discretization used, similarly to our ParaOpt results. Numerical results
confirm that the iteration count of the iterative solvers used for ParaOpt's
linear systems becomes constant in the limit of an increasing processor count.
The paper is accompanied by a sequential MATLAB implementation.
