Extremum Seeking Boundary Control for Euler-Bernoulli Beam PDEs

This paper presents the design and analysis of an extremum seeking (ES) controller for scalar static maps in the context of infinite-dimensional dynamics governed by the 1D Euler-Bernoulli (EB) beam Partial Differential Equation (PDE). The beam is actuated at one end (using position and moment actuators). The map's input is the displacement at the beam's uncontrolled end, which is subject to a sliding boundary condition. Notably, ES for this class of PDEs remains unexplored in the existing literature. To compensate for PDE actuation dynamics, we employ a boundary control law via a backstepping transformation and averaging-based estimates for the gradient and Hessian of the static map to be optimized. This compensation controller leverages a Schr\"odinger equation representation of the EB beam and adapts existing backstepping designs to stabilize the beam. Using the semigroup and averaging theory in infinite dimensions, we prove local exponential convergence to a small neighborhood of the unknown optimal point. Finally, simulations illustrate the effectiveness of the design in optimizing the unknown static map.