Gradient extremum seeking for static maps with actuation dynamics governed by diffusion PDEs

We design and analyze the scalar gradient extremum seeking control feedback for static maps with actuation dynamics governed by diffusion PDEs. Conceptually, a non-model based online optimization control scheme is paired with actuation dynamics which occur in chemistry, biology and economics. A learning-based adaptive control approach with known actuation dynamics is considered in this paper. In the design part, we first compensate the actuation dynamics in the dither signals. Secondly, we introduce an average-based actuation dynamics compensation controller via a backstepping transformation, which is fed by the perturbation-based gradient and Hessian estimates of the static map. The stability analysis of the error-dynamics is based on using Lyapunov's method and applying averaging for infinite-dimensional systems to capture the infinite-dimensional state of the actuator model. Local exponential convergence to a small neighborhood of the optimal point is proven and illustrated by numerical simulations.