Boundary port variables and uniform controllability: the shallow water example

First the classical, port-Hamiltonian and discrete lattice Boltzmann (LBM) formulations for the shallow water equations are presented. Boundary controllability is studied in nonlinear and linearized cases. The controllability of the shallow water equations and the uniform controllability of the LBM are then investigated in the linearized case. It is shown that, using water levels/flows boundaries input variables in the LBM, there is a loss of controllability in the sequence of finite dimensional approximations when the order of the model increases. It is finally shown that we don't have this loss of controllability when the chosen input are boundary port-variables.