Stabilization boundary control and stability analysis for the Burgers' cellular automaton and the difference Burgers' equation based on ultradiscretization

The main purpose of this paper is to develop stabilization boundary control methods for the Burgers' cellular automaton and the difference Burgers' equation based on the ultradiscretization technique, and show stability analysis. We first derive a stabilizing boundary control law for the Burgers' cellular automaton, which is obtained by ultradiscretizing the Burgers' equation, and prove finite stabilization for the controlled Burgers' cellular automaton. Then, by using the inverse ultradiscretization method, we derive a stabilizing boundary control law for the difference Burgers' equation, and show stability and convergence values for the controlled difference Burgers' equation. From some numerical simulations, it turns out that the difference Burgers' equation is stabilized and the convergence value coincides with the theoretical one.