Controlling the dynamics of Burgers equation with a high-order nonlinearity
Summary: We investigate analytically as well as numerically Burgers equation with a high-order nonlinearity (i.e., \(u_{t}=\nu u_{xx}-u^{n}u_{x}+mu+h(x)\)). We show existence of an absorbing ball in \(L^{2}[0,1]\) and uniqueness of steady state solutions for all integer \(n \geq 1\). 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 the Chebychev collocation method with the 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.
scientific article; zbMATH DE number 2171782