Estimation for Linear and Semi-linear Infinite-dimensional Systems

Estimating the state of a system that is not fully known or that is exposed to noise has been an intensely studied problem in recent mathematical history. Such systems are often modelled by either ordinary differential equations, which evolve in finite-dimensional state-spaces, or partial differential equations, the state-space of which is infinite-dimensional. The Kalman filter is a minimal mean squared error estimator for linear finite-dimensional and linear infinite-dimensional systems disturbed by Wiener processes, which are stochastic processes representing the noise. For nonlinear finite-dimensional systems the extended Kalman filter is a widely used extension thereof which relies on linearization of the system. In all cases the Kalman filter consists of a differential or integral equation coupled with a Riccati equation, which is an equation that determines the optimal estimator gain. This thesis proposes an estimator for semi-linear infinite-dimensional systems. It is shown that under some conditions such a system can also be coupled with a Riccati equation. To motivate this result, the Kalman filter for finite-dimensional and infinite-dimensional systems is reviewed, as well as the corresponding theory for both stochastic processes and infinite-dimensional systems. Important results concerning the infinite-dimensional Riccati equation are outlined and existence of solutions for a class of semi-linear infinite-dimensional systems is established. Finally the well-posedness of the coupling between a semi-linear infinite-dimensional system with a Riccati equation is proven using a fixed point argument