Predictor-like feedback for 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.