The root locus is an important tool for analysing the stability and time
constants of linear finite-dimensional systems as a parameter, often the gain,
is varied. However, many systems are modelled by partial differential equations
or delay equations. These systems evolve on an infinite-dimensional space and
their transfer functions are not rational. In this paper a rigorous definition
of the root locus for infinite-dimensional systems is given and it is shown
that the root locus is well-defined for a large class of infinite-dimensional
systems. As for finite-dimensional systems, any limit point of a branch of the
root locus is a zero. However, the asymptotic behaviour can be quite different
from that for finite-dimensional systems. This point is illustrated with a
number of examples. It is shown that the familiar pole-zero interlacing
property for collocated systems with a Hermitian state matrix extends to
infinite-dimensional systems with self-adjoint generator. This interlacing
property is also shown to hold for collocated systems with a skew-adjoint
generator