Observer-based stabilization of an unstable parabolic PDE using the pseudospectral method and Sturm-Liouville theory

The stabilization of an unstable linear parabolic partial differential equation (PDE) system with both Neumann boundary control and interior control is considered in this work. Point output measurement is available at one end of the physical domain. The choice of a proportional output feedback boundary control is justified by Lyapunov analysis while the design of the interior control is carried out based on the Sturm-Liouville theory. A proportional state feedback is proposed for the interior control with a symmetric kernel function, and the pseudospectral method is used to solve the stability conditions governed by the Sturm-Liouville systems. In addition, an observer is designed using the point measurement at one end of the physical domain, and used to propose an observer-based feedback controller for the PDE system. Both controller and observer gains are designed numerically to make the eigenvalues of the associated Sturm-Liouville problems stable. Simulations show the effectiveness of the proposed controller.