Recovering the initial state of an infinite-dimensional system using observers

Let A be the generator of a strongly continuous semigroup T on the Hilbert space X , and let C be a linear operator from D ( A ) to another Hilbert space Y (possibly unbounded with respect to X , not necessarily admissible). We consider the problem of estimating the initial state z 0 ∈ D ( A ) (with respect to the norm of X ) from the output function y ( t ) = C T t z 0 , given for all t in a bounded interval [ 0 , τ ] . We introduce the concepts of estimatability and backward estimatability for ( A , C ) (in a more general way than currently available in the literature), we introduce forward and backward observers, and we provide an iterative algorithm for estimating z 0 from y . This algorithm generalizes various algorithms proposed recently for specific classes of systems and it is an attractive alternative to methods based on inverting the Gramian. Our results lead also to a very general formulation of Russell’s principle, i.e., estimatability and backward estimatability imply exact observability. This general formulation of the principle does not require T to be invertible. We illustrate our estimation algorithms on systems described by wave and Schrödinger equations, and we provide results from numerical simulations.