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-conforming finite element method in spatial direction, optimal error estimates in L^infty (L^2) and in L^infty (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.
Highlights
We consider the following Neumann boundary control problem for the two-dimensional viscous Burgers’ or Bateman–Burgers equation: seek u = u(x, t), t > 0 which satisfies ut − ν u + u(∇u · 1) = 0 in (x, t) ∈ × (0, ∞), (1.1) ∂ ∂ u n (x, t ) = v2on (x, t) ∈ ∂ × (0, ∞), (1.2)u(x, 0) = u0(x) in x ∈
For existence and uniqueness with continuous dependence property of one dimensional Burgers’ equation with similar type nonlinearity, see, [17,22] and their arguments can be modified to prove the wellposedness of the problem (2.14)
Additional regularity results are established assuming compatibility conditions, which are crucial for proving optimal error estimates for the state variable
Summary
Local stabilization result for 2D viscous Burgers’ equation is available in [29] where a nonlinear feedback control law is applied which is obtained through solving Hamilton–. Our attempt in this paper is to establish global stabilization result without smallness assumption on the data through the nonlinear Neumann control law using Lyapunov type functional. Such global stabilization results for one dimensional Burgers’ equation was earlier studied in [3,18] for both Dirichlet and Neumann boundary control laws. With the help of Lyapunov functional, a nonlinear Neumann feedback control law for the problem (1.1)–(1.3) is derived and global stabilization results in L∞(H i ) (i = 0, 1, 2) norms are established.
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