Boundary stabilization of the inviscid Burgers equation using a Lyapunov method

We consider the problem of stabilization of the inviscid Burgers partial differential equation (PDE) using boundary actuation. We propose a solution to the problem using a Lyapunov approach and prove that the inviscid Burgers equation is stabilizable around a constant uniform state under an appropriate boundary control. We conduct this study in the space of weak solutions of the PDE. Because of the absence of viscosity term, discontinuities can appear in finite time for general initial conditions. In order to handle this feature of the solutions, we decompose the Lyapunov function into a sum of functions which can be studied via classical methods. The consideration of weak boundary conditions, common in the field of conservation laws, enables the definition of a control for which the actuator has an effective action. Under the assumption that the solution can be expressed as a finite sum of continuously differentiable functions, we prove that the system is stabilizable in the sense of Lyapunov in the control space of strong boundary conditions. We illustrate the results with numerical simulations based on the Godunov scheme.