Exponential Tracking and Disturbance Rejection for Nonlinear Reaction Convection Diffusion Equations via Boundary Control

Abstract

This paper studies the problem of exponential tracking and disturbance rejection for nonlinear reaction convection diffusion equations via Dirichlet or Neumann boundary control. Disturbances are present both in the equation and on the boundary. The tracking problem is simply a nonlinear dynamic regulator problem. If an initial condition is not compatible with a reference, the nonlinear dynamic regulator problem does not have a solution. Thus, we split the tracking problem into two problems to relax the compatibility condition. We choose one problem to be exponentially stable and leave everything else in a dynamic regulator problem. Then a controller is just a solution component of the dynamic regulator problem, while the tracking error is given by the solution of the exponentially stable problem. The controller can be obtained by integrating the dynamic regulator equation and using the reference condition. The dynamic regulator problem with the obtained controller becomes an initial boundary value problem. We then transfer the initial boundary value problem into an equivalent integral equation and prove that the integral equation has a unique global solution if the nonlinearity is globally Lipschitz. In addition, a numerical example is given to illustrate the theoretical results.