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, applying $$C^0$$ C 0 -conforming finite element method in spatial direction, optimal error estimates in $$L^\infty (L^2)$$ L ∞ ( L 2 ) and in $$L^\infty (H^1)$$ 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.

Local stabilization of viscous Burgers equation with memory

“…In 2000, by introducing a cubic Neumann boundary feedback control, Balogh and Krstic [7] prove global asymptotic stability. In [28][29][30], global stabilization results by using Lyapunov functional are established for Burgers equation, and BBM Burgers equation y t − η 1 y xxt − η 2 y xx + y x + yy x = 0 for all (x, t) ∈ (0, 1) × (0, ∞), y x (0, t) = u 0 (t) for all t ∈ (0, ∞), y x (1, t) = u 1 (t) for all t ∈ (0, ∞), y(x, 0) = y 0 (x) for all x ∈ Ω, where η 1 , η 2 > 0 are given constants, u 0 (t) and u 1 (t) are controls. In these articles, the authors also study error estimates for the stabilized system and verify their results by providing numerical examples.…”

“…They numerically show that the non-linear system is stabilizable by the computed feedback control. Global stabilization results by using Lyapunov functional are established for Burgers equation, and BBM Burgers equation with Neumann boundary control in [28][29][30]. In these articles, the error estimates for the stabilized system are established and verified by providing some numerical examples.…”