Adaptive boundary concentration control using Zakai equation

A mean-variance control problem is formulated with respect to a partially observed nonlinear system that includes unknown constant parameters. A physical prototype of the system is the cathode surface reaction in an electrolysis cell, where the controller aim is to keep the boundary concentration of species in the near vicinity of the cathode surface low but not zero. The boundary concentration is a diffusion-controlled process observed through the measured current density and, in practice, controlled through the applied voltage. The former incomplete data control problem is converted to complete data–to the so-called separated control problem whose solution is given by the infinite-dimensional Zakai equation. In this article, the separated control problem is solved numerically using pathwise integration of the Zakai equation. This article demonstrates precise tracking of the target trajectory with a rapid convergence of estimates to unknown parameters, which take place simultaneously with control.