  We consider the quantum state control of a multi-state system which evolves
an initial state into a target state. We explicitly demonstrate the control
method in an interesting case involving the transfer and rotation of a
Schr\"{o}dinger cat state through a coupled harmonic oscillator chain at a
predetermined time $T$. We use the gradient-based Krotov's method to design the
time-dependent parameters of the coupled chain to find an optimal control shape
that will evolve the system into a target state. We show that the prescribed
quantum state control can be achieved with high fidelity, and the robustness of
the control against generic environment noises is explored. Our findings will
be of interest for the optimal control of a many-body open quantum system in
the presence of environmental noise.
