  We demonstrate that arbitrary time evolutions of many-body quantum systems
can be reversed even in cases when only part of the Hamiltonian can be
controlled. The reversed dynamics obtained via optimal control --contrary to
standard time-reversal procedures-- is extremely robust to external sources of
noise. We provide a lower bound on the control complexity of a many-body
quantum dynamics in terms of the dimension of the manifold supporting it,
elucidating the role played by integrability in this context.
