  We consider a qubit governed by a sequence of weak measurements, with the
measurement strength modified in a time- and state-dependent manner. The
resulting trajectory of the qubit in the phase space can be weakly controlled
without any direct action on the qubit (control-free control), even only one
fixed observable is measured. Here we show a possibility of a weak form of a
stochastic ratchet, allowing to create an additional "force" without changing
the corresponding effective average potential. Furthermore, if the weak
measurement strength is significantly reduced in a way conditioned to some
particular state, a dynamical localization near this state takes place. If the
measurement strength is reduced to zero, a singularity appears, which behaves
like an artificial basis state.
