  We theoretically study the topology of the phase diagram of a family of
quantum models inspired by the classical Bernoulli map under stochastic
control. The quantum models inherit a control-induced phase transition from the
classical model and also manifest an entanglement phase transition intrinsic to
the quantum setting. This measurement-induced phase transition has been shown
in various settings to either coincide or split off from the control
transition, but a systematic understanding of the necessary and sufficient
conditions for the two transitions to coincide in this case has so far been
lacking. In this work, we generalize the control map to allow for either local
or global control action. While this does not affect the classical aspects of
the control transition that is described by a random walk, it significantly
influences the quantum dynamics, leading to the universality class of the
measurement-induced transition being dependent on the locality of the control
operation. In the presence of a global control map, the two transitions
coincide and the control-induced phase transition dominates the
measurement-induced phase transition. Contrarily, the two transitions split in
the presence of the local control map or additional projective measurements and
generically take on distinct universality classes. For local control, the
measurement-induced phase transition recovers the Haar logarithmic conformal
field theory universality class found in feedback-free models. However, for
global control, a novel universality class with correlation length exponent
$\nu \approx 0.7$ emerges from the interplay of control and projective
measurements. This work provides a more refined understanding of the
relationship between the control- and measurement-induced phase transitions.
