Well-posedness of Extended Kalman Filter equations for semilinear infinite-dimensional systems

In many physical applications, the system's state varies with spatial variables as well as time. The state of such systems evolves on an infinite-dimensional space, and so they are an important class of infinite-dimensional systems. 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 is one type of extension to nonlinear finite-dimensional systems. In this paper we provide first steps of an extension of the extended Kalman filter to semilinear infinite-dimensional systems. Under mild assumptions we prove the well-posedness of equations defining the EKF. Simulations illustrate the efficacy of the observer.