Nonlinear stabilization of shock-like unstable equilibria in the viscous burgers PDE

We stabilize the unstable "shock-like" equilibrium profiles of the viscous Burgers equation using control at the boundaries. These equilibria are not stabilizable (even locally) using the standard "radiation feedback boundary conditions." Using a nonlinear spatially-scaled transformation (that employs three ingredients, of which one is the Hopf--Cole non-linear integral transformation) and linear backstepping, we design an explicit nonlinear full-state control law that achieves exponential stability, with a region of attraction for which we give an estimate. The region of attraction is not the entire state space since the Burgers PDE is known not to be globally controllable. Index Terms--Burgers PDE.