Extremum seeking boundary control for PDE–PDE cascades

Abstract The goal of this paper is to shed light on new opportunities for extremum seeking through extensions to locally quadratic nonlinear maps with actuator dynamics modeled by cascades of partial differential equations (PDEs). First, we deal with PDEs with input delays such as, for example, the notoriously difficult problem of a wave PDE with input delay where, if the delay is left uncompensated, an arbitrarily short delay destroys the closed-loop stability. Then, we move forward to an even more challenging class of problems for parabolic–hyperbolic cascades of PDEs, coping with a heat equation at the input of a wave PDE. The treatment of such systems with PDE–PDE cascades is performed by means of boundary control. The proposed approach yields (small-gain) loops that make the control design constructive and enables stability analysis with quantitative estimates. Local exponential stability and convergence to a small neighborhood of the unknown extremum point are rigorously guaranteed. This result is achieved by using backstepping transformation and averaging in infinite dimensions. Although we keep the presentation using the classical Gradient extremum seeking, the generalization of the results for the Newton-based approach is also indicated. A numerical example is given to illustrate the effectiveness of the proposed extremum seeking boundary control for compensation of PDE–PDE cascades.