Dynamics of boundary-controlled convective reaction-diffusion equations

This research is concerned with an initial value boundary problem for a class of convective reaction-diffusion equations for which a feedback control law is implemented through the boundary conditions. This class contains, as a special case the well-known Burgers' equation which has been studied rather extensively. Using methods based on Functional Analysis, in particular, the energy method and Galerkin Approximations, solvability for the above class is established. In addition, we prove the global in time existence and the regularity of solutions of the controlled problem for sufficiently small L^2-initial data. To do this, additional explicit restrictions on the nonlinear terms are imposed. Then we prove the local Lyapunov stability of the system, the existence of an absorbing ball, and the existence of a compact local attractor in this ball. Similar results for the same equation with Dirichlet boundary conditions are obtained for arbitrary L^2-initial data.

The solutions of the boundary-controlled problem are shown to depend continuously on the boundary control parameters. As these parameters tend to infinity, we prove that the trajectories of the boundary-controlled problem converge, uniformly on any finite interval, to the trajectories of the corresponding problem with Dirichlet boundary conditions.