Nonlinear Local Stabilization of a Viscous Hamilton-Jacobi PDE

We consider the boundary stabilization problem of the non-uniform equilibrium profiles of a viscous Hamilton-Jacobi (HJ) Partial Differential Equation (PDE) with parabolic concave Hamiltonian. We design a nonlinear full-state feedback control law, assuming Neumann actuation, which achieves an arbitrary rate of convergence to the equilibrium. Our design is based on a feedback linearizing transformation which is locally invertible. We prove local exponential stability of the closed-loop system in the H^1 norm, by constructing a Lyapunov functional, and provide an estimate of the region of attraction. We design an observer-based output-feedback control law, by constructing a nonlinear observer, using only boundary measurements. We illustrate the results on a benchmark example computed numerically.