Predictive Control with guaranteed stability for hyperbolic systems of conservation laws

This paper deals with the Predictive Control for a linear hyperbolic system of conservation laws. A complete proof of the exponential stability of this control is established. The semi-group approach is used to prove the existence and uniqueness of the optimal solution. The cost function of the optimal control is inspired from the previously proposed candidate Lyapunov function for hyperbolic systems. Thanks to this choice, the exponential stability of the control is proven. For the implementation, calculus of variations is used to derive the adjoint state of the system and the recently proposed Lattice Boltzmann method is used to solve both direct and adjoint partial differential equations. This approach is finally validated in simulation.