Boundary Geometric Control of a Nonlinear Diffusion System with Time-Dependent Spatial Domain
Résumé
A Stefan problem represents a distributed parameter system with a time-dependent spatial domain. This paper addresses the boundary control of the position of the moving liquid-solid interface in the case of nonlinear Stefan problem with Neumann actuation. The main idea consists in deriving an equivalent linear model by means of Cole-Hopf tangent transformation, i.e. under a certain physical assumption, the original nonlinear Stefan problem is converted to a linear one. Then, the geometric control law is deduced directly from that developed, by the authors of the present paper, for the linear Stefan problem. Based on the fact that the Cole-Hopf transformation is bijective, it is shown that the developed control law yields a stable closed-loop system. The performance of the controller is evaluated through numerical simulation in the case of stainless steel melting characterized by a temperature-dependent thermal conductivity, which is nonlinear. The objective is to control the position of the liquid-solid interface by manipulating a heat flux at the boundary.
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