Boundary and distributed control of the viscous Burgers equation

In this paper, the dynamics of the forced Burgers equation: u t = ν u xx - uu x + f ( x ) , subject to both Neumann boundary conditions and periodic boundary conditions using boundary and distributed control is analyzed. For the boundary control problem, we show that the controlled unforced Burgers equation (i.e., the closed loop system) is exponentially stable when the viscosity ν is known, and globally asymptotically stable when ν is unknown. As for the distributed control problem, we apply Karhunen–Loéve decomposition on the dynamics of the forced Burgers equation to generate a low dimensional dynamical system whose dynamics is similar to that of Burgers equation. Then, a feedback linearization control is used on the reduced system to exponentially stabilize the dynamics of the equation. Numerical simulations for the boundary and distributed controls are presented to support the analytical results.