Viability approach for the Saint-Venant equations for modeling and controling hydrosystems

The Saint-Venant equations are two coupled partial derivative equations describing the average velocity and the water depth of a water flow circulating in channels for instance for supplying water over large countrysides. The first one is the mass conservation equation:

$$\frac{\partial A(x,t)}{\partial t}+\frac{\partial Q(x,t)}{\partial x} =0,$$

and the second one is the momentum conservation equation:

$$ \frac{\partial Q(x,t)}{\partial t}+\frac{\partial }{\partial x}\Bigl[\frac{Q^{2}(x,t)}{A(x,t)}\Bigr]+gA(x,t)\Bigl(\frac{\partial Y(x,t)}{\partial x}+S_f(x,t)-S_b(x)\Bigr) =0. $$

The solutions of these two PDE equations can be linked to a set of four ordinary differential equations in the \((x,t)\) plane using the classical "characteristic form". From these results, applying viability approach we prove that the Saint-Venant solution satisfying an initial condition \(A(x,0):=A_0(x)\) and \(Q(x,0):=Q_0(x)\) are in relation to the viable capture basin of a target for a particular dynamical system. This viable capture basin coincides with the intersection of two epigraphs of two maps \(\Phi\) and \(\Psi\) that are lower semi-continuous and can be computed using the Capture Basin Algorithm. From \(\Phi\) and \(\Psi\) we derive the solution of the Saint-Venant dynamical system. From this result one can give a sufficient condition for avoiding shocks and moreover this result open the door to water distribution controlling problem in irrigation process in the presence of uncertainty, for instance in weather forecasting.