Traps in control problems of two-level quantum systems

In the report we will discuss the problem of local maxima and minima (traps) of the objective functional for various problems of control of two-level quantum systems. The analysis of traps is of significant practical importance, since traps can make difficult the search of global maximum or minimum of objective functional. We will present two theorems describing trapping behavior in two-level quantum systems. The first states that there are no traps for sufficiently large duration of the control pulse. The second states that for sufficiently short duration of the control pulse the objective functional describing maximization of the probability of transition between pair of states has one trap in a certain region of parameters of the controlled system.