  A general scheme is presented for controlling quantum systems using evolution
driven by non-selective von Neumann measurements, with or without an additional
tailored electromagnetic field. As an example, a 2-level quantum system
controlled by non-selective quantum measurements is considered. The control
goal is to find optimal system observables such that consecutive non-selective
measurement of these observables transforms the system from a given initial
state into a state which maximizes the expected value of a target operator (the
objective). A complete analytical solution is found including explicit
expressions for the optimal measured observables and for the maximal objective
value given any target operator, any initial system density matrix, and any
number of measurements. As an illustration, upper bounds on measurement-induced
population transfer between the ground and the excited states for any number of
measurements are found. The anti-Zeno effect is recovered in the limit of an
infinite number of measurements. In this limit the system becomes completely
controllable. The results establish the degree of control attainable by a
finite number of measurements.
