  Symmetry groups of PDEs allow to transform solutions continuously into other
solutions. In this paper, we use this property for the observability analysis
of nonlinear PDEs with input and output. Based on a differential-geometric
representation of the nonlinear system, we derive conditions for the existence
of special symmetry groups that do not change the trajectories of the input and
the output. If such a symmetry group exists, every solution can be transformed
into other solutions with the same input and output trajectories but different
initial conditions, and this property can be used to prove that the system is
not observable. We also put emphasis on showing how the approach simplifies for
linear systems, and how it is related to the well-known observability concepts
from infinite-dimensional linear systems theory.
