An H2 control performance criterion dependent on a controller parameter vector is analyzed from the point of view of evaluating the domain of attraction of its global minimum when a gradient-based algorithm is used in tuning the parameters. The objective of this analysis is twofold: (i) examine how some design parameters of the criterion can be used to enlarge the domain of attraction; (ii) examine how the minimization of a sequence of intermediate criteria can possibly lead to the global minimum of the original criterion without the danger of entrapment in a local minimum.