Controlling the dynamics of Burgers equation with a high-order nonlinearity

We investigate analytically as well as numerically Burgers equation with a high-order nonlinearity (i.e., ut = νuxx−unux+mu+h(x)). We show existence of an absorbing ball in L2[0,1] and uniqueness of steady state solutions for all integer n ≥ 1. Then, we use an adaptive nonlinear boundary controller to show that it guarantees global asymptotic stability in time and convergence of the solution to the trivial solution. Numerical results using Chebychev collocation method with backward Euler time stepping scheme are presented for both the controlled and the uncontrolled equations illustrating the performance of the controller and supporting the analytical results.