Compensating actuator and sensor dynamics governed by diffusion PDEs

For (possibly unstable) ODE systems with actuator delay, predictor-based infinite-dimensional feedback can compensate for actuator delay of arbitrary length and achieve stabilization. We extend this concept to another class of PDE-ODE cascades, where the infinite-dimensional part of the plant is of diffusive, rather than convective type. We derive predictor-like feedback laws and observers, with explicit gain kernels. The gain kernels involve second-order matrix exponentials of the system matrix of the ODE plant, which is the result of the second-order-in-space character of the actuator/sensor dynamics. The construction of the kernel functions is performed using the continuum version of the backstepping method. Robustness to small perturbations in the diffusion coefficient is proved.