  We find dynamical invariants for open quantum systems described by the
non-Markovian quantum state diffusion (QSD) equation. In stark contrast to
closed systems where the dynamical invariant can be identical to the system
density operator, these dynamical invariants no longer share the equation of
motion for the density operator. Moreover, the invariants obtained with from
bi-orthonormal basis can be used to render an exact solution to the QSD
equation and the corresponding non-Markovian dynamics without using master
equations or numerical simulations. Significantly we show that we can apply
these dynamic invariants to reverse-engineering a Hamiltonian that is capable
of driving the system to the target state, providing a novel way to design
control strategy for open quantum systems.
