Extremum Seeking for a Class of Wave Partial Differential Equations With Kelvin-Voigt Damping

This letter presents the design and analysis of gradient extremum seeking applied to scalar static maps within the context of infinite-dimensional dynamics governed by Partial Differential Equations (PDEs) of wave type featuring a small amount of Kelvin-Voigt damping. Notably, this particular class of PDEs for extremum seeking still needs to be explored in the existing literature. First, to compensate for the influence of PDE actuation dynamics, we employ a boundary control law via backstepping transformation and averaging-based estimates of the gradient and Hessian. Finally, we prove the local exponential convergence to a small neighborhood surrounding the unknown optimal point by means of an Input-to-State Stability analysis, as well as by employing Lyapunov functionals and averaging theory in infinite dimensions.