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

International audienceLet $A$ be the generator of a strongly continuous semigroup ${\mathbb{T}}$ on the Hilbert space $X$, and let $C$ be a linear operator from $\Dscr(A)$ to another Hilbert space $Y$ (possibly unbounded with respect to $X$, not necessarily admissible). We consider the problem of estimating the initial state $z_0\in\Dscr(A)$ (with respect to the norm of $X$) from the output function $y(t)=C{\mathbb{T}}_t z_0$, given for all $t$ in a bounded interval $[0,\tau]$. 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 $\stline$ to be invertible. We illustrate our estimation algorithms on systems described by wave and Schrödinger equations, and we provide results from numerical simulations