The Stability Control And Numerical Analysis Of MKdV-Burgers Equation

This paper studies an important kind of nonlinear evolution equation: MKdV-Burgers equation . Firstly, The equilibrium solution is global asymptotic stability and exponential stability on L2[0,l] under Neumann and Dirichlet boundary control conditions, control inputs for choosing the boundary control is bounded in Lx, the equilibrium solutions decay to zero and integrating the solution square in [0,1], it decays to zero exponentially, by using nonlinear boundary control conditions and input feedback control method. Applying the results to optimal controller to realize the minimizer of the cost function of MKdV-Burgers equation under Neumann and Dirichlet boundary control.Secondly, using inertial manifold and approximate inertial manifold theory,the approximate inertial manifold under Fourier bases is given and we construct a set of ODES of three modes to obtain the long-time dynamic behavior. Numerical result is also given.