Newton-based extremum seeking for a static map coupled with a diffusion PDE at an arbitrary interior point

In this paper the Newton-based extremum seeking for an unknown static map which is coupled with a diffusion partial differential equation (PDE) is presented. In contrast with previous works where optimization takes place at the zero boundary of the diffusion PDE, this paper considers a more general case where the optimization can take place at an arbitrary interior point of the diffusion PDE. First, the additive and multiplicative dither signals are designed to provide the estimations of the gradient and Hessian of the map in the average sense. Then, the error system is formulated as a first-order ordinary differential equation (ODE) coupled with a diffusion PDE at an interior point and an explicit dynamic feedback law is designed for stabilizing it. By using the backstepping transformation and infinite dimensional averaging theorem, it is shown that the proposed scheme achieves local exponential convergence to a neighborhood of the extremum. A simulation example is given to illustrate the theoretical results.