  We present a fast scheme for arbitrary unitary control of interacting bosonic
atoms in a double-well. Assuming fixed inter-well tunnelling rate and
intra-well interaction strength, we control the many-atom state by a discrete
sequence of shifts of the single-well energies. For strong interactions,
resonant tunnelling transitions implement beam-splitter U(2) rotations among
atom number eigenstates, which can be combined and, thus, permit full
controllability. By numerically optimizing such sequences of couplings at
avoided level crossings (CALC), we extend the realm of full controllability to
a wide range of realistic interaction parameters, while we remain in the simple
control space. We demonstrate the efficiency and the high achievable fidelity
of our proposal with non-adiabatic population transfer, N00N-state creation, a
C-NOT gate, and a transistor-like, conditional evolution of several atoms.
