Estimation of the potential in a 1D wave equation via exponential observers

The problem of estimation of the unknown potential in a 1-dimensional wave equation via state observers is considered in this work. The potential is supposed to depend on the space variable only and be polynomial. The main observation information is the value of the solution of the wave equation in a subinterval of the domain, including also some of its higher-order spatial derivatives. The method we propose to estimate the potential includes turning it into a new state as in finite-dimensional parameter estimation approaches. However, in this infinite dimensions setting, this requires an indirect approach that is introduced, including an infinite-dimensional state transformation. Sufficient conditions allow the design of an internal semilinear observer for the resulting cascade system, corresponding to the observed subinterval, which estimates the potential in an exponentially fast manner.