Observers and initial state recovering for a wave equation: an LMI approach *

Recently the problem of estimating the initial state of some linear infinite-dimensional systems from measurements on a finite interval was solved by using the sequence of forward and backward observers [14]. In the present paper, we introduce a direct Lyapunov approach to the problem and extend the results to the class of semilinear systems governed by 1-d wave equations with boundary measurements from a finite interval. We first design forward observers and derive Linear Matrix Inequalities (LMIs) for the exponential stability of the estimation errors. Further we find LMIs for an upper bound T* on the minimal time, that guarantees the convergence of the sequence of forward and backward observers on [0, T*] for the initial state recovering. For observation times bigger than T*, these LMIs give upper bounds on the convergence rate of the iterative algorithm in the norm defined by the Lyapunov functions. The efficiency of the results are illustrated by a numerical example.