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This paper presents the design and analysis of the extremum seeking forstatic maps with input passed through a partial differential equation (PDE) ofthe diffusion type defined on a time-varying spatial domain whose boundaryposition is governed by an ordinary differential equation (ODE). This is thefirst effort to pursue an extension of extremum seeking from the heat PDE tothe Stefan PDE. We compensate the average-based actuation dynamics by acontroller via backstepping transformation for the moving boundary, which isutilized to transform the original coupled PDE-ODE into a target system whoseexponential stability of the average equilibrium of the average system isproved. The discussion for the delay-compensated extremum seeking control ofthe Stefan problem is also presented and illustrated with numericalsimulations.

Extremum Seeking for Stefan PDE with Moving Boundary