  We present a comprehensive analysis of the landscape for full quantum-quantum
control associated with the expectation value of an arbitrary observable of one
quantum system controlled by another quantum system. It is shown that such full
quantum-quantum control landscapes are convex, and hence devoid of local
suboptima and saddle points that may exist in landscapes for quantum systems
controlled by time-dependent classical fields. There is no controllability
requirement for the full quantum-quantum landscape to be trap-free, although
the forms of Hamiltonians, the flexibility in choosing initial state of the
controller, as well as the control duration, can infulence the reachable
optimal value on the landscape. All level sets of the full quantum-quantum
landscape are connected convex sets. Finally, we show that the optimal solution
of the full quantum-quantum control landscape can be readily determined
numerically, which is demonstrated using the Jaynes-Cummings model depicting a
two-level atom interacting with a quantized radiation field.
