Existence of traps in the problem of maximizing quantum observable averages for a qubit at short times

We consider the Mayer maximization problem for an objective functional that describes the average value at some fixed time of a quantum-mechanical observable for a two-level quantum system (qubit). In the previous studies we proved that for sufficiently large times the objective functional has no local maxima that are not global maxima. Such local maxima that are not global are called traps. In this paper we prove that for sufficiently short times under certain conditions traps for this problem do exist.