Boundary Control of the Forced Viscous Burgers Equation

In this paper, the control problem of the forced Burgers equation: ∂u ∂t = ν ∂2u ∂x2 − u ∂u ∂x +mu+ f(x), 0 < x < 2π, t > 0 subject to Neumann boundary conditions: ∂u ∂x (0, t) = ũ1(t), ∂u ∂x (2π, t) = ũ2(t) t > 0. and the initial condition: u(x, 0) = u0(x), x ∈ (0, 2π) t > 0, where ν is a positive constant, the parameter m ∈ IR, and ũ1(t), ũ2(t) are two control inputs is considered. We show that the controlled forced Burgers equation is exponentially stable when f ∈ L2(0, 2π), and the viscosity ν > 4π2(2m+ 1). AMS Subject Classification: 34H05, 35B37, 35Q53, 93D15