On the Use of a Global Control Lyapunov Functional in Infinite-dimensional Predictive Control
Résumé
The paper introduces a Global Control Lyapunov Functional (GCLF) which is an extension of the Lyapunov function proposed by Coron et al. [2007] for linear systems of conservation laws. This function has the advantage in comparison with the one of Coron et al. [2007] that it can deal with all initial conditions. From this, it can be used in the context of the Receding horizon optimal control, and - combined with the approach used by Findensen and Allgower [2002] - to prove that the minimum of the cost function decreases along the corresponding controlled trajectories. Furthermore, by exploiting the argument of Ito and Funisch [2002], it can be proven that this function decreases exponentially. All those results are illustrated by simulations in the case of an open hydraulic channel which is described by Saint-Venant equations. For the simulation, calculus of variations is used to derive the adjoint state of the system and the recently proposed Lattice Boltzmann method (see Zhou [2004], Chopard et al. [2009], Junk and Rheinlainder [2008]) is used to solve both direct and adjoint partial differential equations.