  In the spirit of the well-known odd-number limitation, we study failure of
Pyragas control of periodic orbits and equilibria. Addressing the periodic
orbits first, we derive a fundamental observation on the invariance of the
geometric multiplicity of the trivial Floquet multiplier. This observation
leads to a clear and unifying understanding of the odd-number limitation, both
in the autonomous and the non-autonomous setting. Since the presence of the
trivial Floquet multiplier governs the possibility of successful stabilization,
we refer to this multiplier as the determining center. The geometric invariance
of the determining center also leads to a necessary condition on the gain
matrix for the control to be successful. In particular, we exclude scalar
gains. Application of Pyragas control on equilibria does not only imply a
geometric invariance of the determining center, but surprisingly also on
centers which resonate with the time delay. Consequently, we formulate odd- and
any-number limitations both for real eigenvalues together with arbitrary time
delay as well as for complex conjugated eigenvalue pairs together with a
resonating time delay. The very general nature of our results allows for
various applications.
