  We address the problem of dynamic output feedback stabilization at an
unobservable target point. The challenge lies in according the antagonistic
nature of the objective and the properties of the system: the system tends to
be less observable as it approaches the target. We illustrate two main ideas:
well chosen perturbations of a state feedback law can yield new observability
properties of the closed-loop system, and embedding systems into bilinear
systems admitting observers with dissipative error systems allows to mitigate
the observability issues. We apply them on a case of systems with linear
dynamics and nonlinear observation map and make use of an ad hoc
finite-dimensional embedding. More generally, we introduce a new strategy based
on infinite-dimensional unitary embeddings. To do so, we extend the usual
definition of dynamic output feedback stabilization in order to allow
infinite-dimensional observers fed by the output. We show how this technique,
based on representation theory, may be applied to achieve output feedback
stabilization at an unobservable target.
