  Whether P systems with only one catalyst can already be computationally
complete, is still an open problem. Here we establish computational
completeness by using specific variants of additional control mechanisms. At
each step using only multiset rewriting rules from one set of a finite number
of sets of multiset rewriting rules allows for obtaining computational
completeness with one catalyst and only one membrane. If the targets are used
for choosing the multiset of rules to be applied, for getting computational
completeness with only one catalyst more than one membrane is needed. If the
available sets of rules change periodically with time, computational
completeness can be obtained with one catalyst in one membrane. Moreover, we
also improve existing computational completeness results for P systems with
mobile catalysts and for P systems with membrane creation.
