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

Let AA be the generator of a strongly continuous semigroup TT on the Hilbert space XX, and let CC be a linear operator from D(A)D(A) to another Hilbert space YY (possibly unbounded with respect to XX, not necessarily admissible). We consider the problem of estimating the initial state z0∈D(A)z0∈D(A) (with respect to the norm of XX) from the output function y(t)=CTtz0y(t)=CTtz0, given for all tt in a bounded interval [0,τ][0,τ]. We introduce the concepts of estimatability and backward estimatability for (A,C)(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 z0z0 from yy. 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 TT to be invertible. We illustrate our estimation algorithms on systems described by wave and Schrödinger equations, and we provide results from numerical simulations.