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 1⁡(⋅, ) of the unknown part in the PDE model depends on the spatial variable and 1⁡(⋅, )∈ 2⁡(0,1).