Fredholm Backstepping Control of Coupled Linear Parabolic PDEs With Input and Output Delays

This paper considers the observer-based output feedback stabilization of coupled parabolic PDEs with spatially-varying coefficients and different diffusion coefficients subject to distinct constant input and output delays. By representing the delays in the form of homodirectional hyperbolic systems, a hyperbolic-parabolic PDE-PDE-PDE cascade is obtained. The state feedback controller design for this system is based on a Fredholm backstepping transformation mapping the closed-loop system into a stable PDE-PDE-PDE target system. With this, a minimum control time is achieved for the hyperbolic subsystem in the target dynamics. Then, this approach is extended to the design of a state observer for delayed measurements to obtain an observer-based output feedback controller. Exponential stability of the resulting closed-loop system is verified. Furthermore, it is shown that the Fredholm and backstepping transformations related to the hyperbolic subsystems are attainable in closed form. The new design procedure is demonstrated for the output feedback stabilization of two coupled parabolic PDEs with different input and output delays.