Adaptive boundary observer for parabolic PDEs subject to domain and boundary parameter uncertainties

We are considering the problem of state observation for a class of infinite dimensional systems modeled by parabolic type PDEs. The model is subject to parametric uncertainty entering in both the domain equation and the boundary condition. An adaptive boundary observer, providing online estimates of the system state and parameters, is designed using finite- and infinite-dimensional backstepping-like transformations. The observer is exponentially convergent under an ad hoc persistent excitation condition.