  There have been recent efforts that combine seemingly disparate methods,
extremum seeking (ES) optimization and partial differential equation (PDE)
backstepping, to address the problem of model-free optimization with PDE
actuator dynamics. In contrast to prior PDE-compensating ES designs, which only
guarantee local stability around the extremum, we introduce unbiased ES that
compensates for delay and diffusion PDE dynamics while ensuring exponential and
unbiased convergence to the optimum. Our method leverages exponentially
decaying/growing signals within the modulation/demodulation stages and
carefully selected design parameters. The stability analysis of our designs
relies on a state transformation, infinite-dimensional averaging, local
exponential stability of the averaged system, local stability of the
transformed system, and local exponential stability of the original system.
Numerical simulations are presented to demonstrate the efficacy of the
developed designs.
