Adaptive Observer for Coupled Wave PDE and Infinite ODE With Sampled Data and Unknown Input: Application to Brain Hemodynamics Estimation

In this article, we investigate the joint estimation problem of the states and the in‐domain Partial Differential Equation (PDE) parameters along with the unknown time‐varying input for coupled wave PDE and infinite‐dimensional Ordinary Differential Equation (ODE) systems. A finite set of measurements described by the sampled‐in‐space state of the ODE system is available to recover the distributed states along with the unknown parameters and input. Through the proposition of an appropriate Lyapunov functional the convergence proof is shown. The proposed Lyapunov functional considers lifted state solutions belonging to more regularized Hilbert spaces. Sufficient conditions on the observer's and adaptation laws' gains, along with the measurements' maximum sampling interval, are derived to guarantee the practical convergence of state estimation errors (uniform ultimate boundedness of the state estimation errors). The proposed observer is applied to characterize the spatiotemporal hemodynamic response in the brain from voxel‐wise functional magnetic resonance imaging (fMRI) measurements. Numerical simulations are provided to demonstrate the efficiency of the theoretical findings.