Adaptive output feedback control for complex-valued reaction-advection-diffusion systems

We study a problem of output feedback stabilization of complex-valued reaction-advection-diffusion systems with parametric uncertainties (these systems can also be viewed as coupled parabolic PDEs). Both sensing and actuation are performed at the boundary of the PDE domain and the unknown parameters are allowed to be spatially varying. First, we transform the original system into the form where unknown functional parameters multiply the output, which can be viewed as a PDE analog of observer canonical form. Input and output filters are then introduced to convert a dynamic parametrization of the problem into a static parametrization where a gradient estimation algorithm is used. The control gain is obtained by solving a simple complex-valued integral equation online. The solution of the closed-loop system is shown to be bounded and asymptotically stable around the zero equilibrium. The results are illustrated by simulations.