  This paper deals with the gradient extremum seeking control for static scalar
maps with actuators governed by distributed diffusion partial differential
equations (PDEs). To achieve the real-time optimization objective, we design a
compensation controller for the distributed diffusion PDE via backstepping
transformation in infinite dimensions. A further contribution of this paper is
the appropriate motion planning design of the so-called probing (or
perturbation) signal, which is more involved than in the non-distributed
counterpart. Hence, with these two design ingredients, we provide an
averaging-based methodology that can be implemented using the gradient and
Hessian estimates. Local exponential stability for the closed-loop equilibrium
of the average error dynamics is guaranteed through a Lyapunov-based analysis.
By employing the averaging theory for infinite-dimensional systems, we prove
that the trajectory converges to a small neighborhood surrounding the optimal
point. The effectiveness of the proposed extremum seeking controller for
distributed diffusion PDEs in cascade of nonlinear maps to be optimized is
illustrated by means of numerical simulations.
