Global Stabilization of Two Dimensional Viscous Burgers' Equation by Nonlinear Neumann Boundary Feedback Control and its Finite Element Analysis

Abstract

In this article, global stabilization results for the two dimensional viscous Burgers' equation, that is, convergence of unsteady solution to its constant steady state solution with any initial data, are established using a nonlinear Neumann boundary feedback control law. Then, using C 0 -conforming finite element method in spatial direction, global stabilization results for the semidiscrete solution are shown. Moreover, optimal error estimates in L ∞ (L 2 ) and in L ∞ (H 1 )norms for the state variable and convergence result for the boundary feedback control law are derived. All the results preserve exponential stabilization property. Finally, several numerical experiments are conducted to confirm our theoretical findings.

Global Stabilization of 2D Forced Viscous Burgers' Equation Around Nonconstant Steady State Solution by Nonlinear Neumann Boundary Feedback Control:Theory and Finite Element Analysis

“…for each y ∈ H 1 (Ω). Below, we recall the following inequalities for our subsequent use Friedrichs's inequality: (See [15]) For y ∈ H 1 (Ω), there holds…”

“…In addition, for proving error estimates for state variable and feedback controllers, we need the following estimate of η and η t at boundary which are proved in [15]. Lemma 4.8.…”

“…Finally, optimal error estimates for the state variable and superconvergence result for the control laws are established. This analysis is then extended to 2D Burgers' equation in [15], and stabilization results depicting convergence of the solution to its constant steady state solution are derived. Moreover, convergence result for the nonlinear feedback control law is also documented.…”