Extremum seeking for infinite-dimensional systems

The last year was the centennial of the 1922 invention, in the context of maximizing the power transfer to an electric tram car, of the method for real-time model-free optimization called extremum seeking (ES). Forgotten after a few decades of practical use and attempts at theoretical study, ES returned to life when the stability of its operation for not only static input–output maps but for dynamic systems modeled by general nonlinear ordinary differential equations (ODEs) was proven in 1997. Then it is natural to ask “why limit the use and the theoretical advances of ES to ODE systems?” So many—if not the majority of physical systems—involve delays or are modeled by partial differential equations (PDEs); why not pursue the application of ES in the presence of delays and for PDE systems? With this tutorial paper, we show a small portion of the vast space of possibilities of designing ES algorithms for infinite-dimensional systems governed by transport hyperbolic PDEs as well as diffusion parabolic PDEs.