  The evolution problem for a quantum particle confined in a 1D box and
interacting with one fixed point through a time dependent point interaction is
considered. Under suitable assumptions of regularity for the time profile of
the Hamiltonian, we prove the existence of strict solutions to the
corresponding Schr\"odinger equation. The result is used to discuss the
stability and the steady-state local controllability of the wavefunction when
the strenght of the interaction is used as a control parameter.
