Nonlinear Control of the Burgers PDE—Part I: Full-State Stabilization

We consider the problem of stabilization of unstable “shock-like” equilibrium profiles of the viscous Burgers’ equation with actuation at the boundaries. These equilibria are not stabilizable (even locally) using the standard “radiation boundary condition.” Using a nonlinear spatially-scaled transformation (that employs three ingredients, of which one is the Hopf-Cole nonlinear integral transformation) and linear backstepping design, 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 equation is known not to be globally controllable, however, the stability result achieved is stronger than being infinitesimally local. In a companion paper we consider output feedback stabilization, for which we design a nonlinear observer with boundary sensing, and solve the problems of trajectory generation and tracking.