Finite-dimensional adaptive observer design for linear parabolic systems with delayed measurements

New finite-dimensional adaptive observers are proposed for uncertain heat equation and a class of linear Kuramoto–Sivashinsky equation (KSE) with local output. The observers are based on the modal decomposition approach and use a classical persistent excitation condition to ensure practical exponential convergence of both states and parameters estimation. An important challenge of this work is that it treats the case when the function (Formula presented.) of the unknown part in the PDE model depends on the spatial variable and (Formula presented.).