Controlling Systems of Conservation Laws - The case of the Open Canals

Among various real life models for controlled conservation laws, the dynamics of the open canal water flow is such that the nonlinear terms of the partial differential equations cannot be neglected. Starting from a rather complicated model deduced for a quite general case, there are considered some simpler models of the practice, relying on the model of the horizontal prismatic canal with simple cross-section (rectangular, trapezoidal). Basic theory (existence, uniqueness, continuous data dependence) is discussed for small deviations around a steady state. There is also discussed the problem of the invariant sets accounting for fluvialness - some kind of regular behavior of the flow reflected in the partial differential equations and their solutions: absence of cavitation, shock and/or rarefaction waves. Based on the Riemann invariants of the equations there is made the synthesis of the feedback structure for the controllers. These are nonlinear decentralized small gain controllers ensuring linearization and dissipativeness of the boundary conditions in their Riemann invariant forms.