Prescribed-Time Extremum Seeking with Chirpy Probing for PDEs—Part II: Heat PDE

We introduce a prescribed–time extremum seeking (PT-ES) design for a PDE-ODE cascade of a heat PDE feeding into an integrator, which in turn feeds into an unknown map. Leveraging the integrator in the PDE-ODE plant, and employing “chirpy” probing and demodulation signals designed by PDE motion planning methods, we achieve convergence to the extremum in a user-prescribed time independent of the distance of the initial estimate from the optimizer. Although this PDE-ODE cascade is defined on a fixed spatial domain, it is inspired by free boundary models such as the Stefan model of phase change dynamics. The design is based on the time-varying backstepping approach, which transforms the PDE-ODE cascade into a suitable prescribed-time stable target system, and the averaging-based estimations of the gradient as well as the Hessian of the map. By means of Lyapunov method, it is shown that the average closed-loop dynamics are prescribed-time stable. This Part II paper is companion to a Part I paper which introduces PT-ES for two problems that are less challenging than here: a static map and a map with an input delay.