Observers and initial state recovering for a class of hyperbolic systems via Lyapunov method

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 Ramdani, Tucsnak, and Weiss (2010). 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 wave and beam 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 obtain simple finite-dimensional conditions in terms of LMIs for an upper bound T∗T∗ on the minimal time, that guarantees the convergence of the sequence of forward and backward observers on [0,T∗][0,T∗] for the initial state recovering. This T∗T∗ represents also an upper bound on the observability time. For observation times bigger than T∗T∗, these LMIs give upper bounds on the convergence rate of the iterative algorithm in the norm defined by the Lyapunov functions. In our approach, T∗T∗ is found as the minimal dwelling time for the switched exponentially stable (forward and backward estimation error) systems with the different Lyapunov functions (Liberzon, 2003). The efficiency of the results is illustrated by numerical examples.