  In many physical applications, the system's state varies with spatial
variables as well as time. The state of such systems is modelled by partial
differential equations and evolves on an infinite-dimensional space. Systems
modelled by delay-differential equations are also infinite-dimensional systems.
The full state of these systems cannot be measured. Observer design is an
important tool for estimating the state from available measurements. For linear
systems, both finite- and infinite-dimensional, the Kalman filter provides an
estimate with minimum-variance on the error, if certain assumptions on the
noise are satisfied. The extended Kalman filter (EKF) is one type of extension
to nonlinear finite-dimensional systems. In this paper we provide an extension
of the EKF to semilinear infinite-dimensional systems. Under mild assumptions
we prove the well-posedness of equations defining the EKF. Local exponential
stability of the error dynamics is shown. Only detectability is assumed, not
observability, so this result is new even for finite-dimensional systems. The
results are illustrated with implementation of finite-dimensional
approximations of the infinite-dimensional EKF on an example.
