Observer-based data assimilation for shallow water equations

We consider state estimation for a system described by the one-dimensional shallow water equations. Since in general measurements of the complete state are not available, we assume that we have measurements of only one state variable, e.g. of the water height. In order to estimate the system state from these partial measurements, we construct an observer system that is based on the shallow water equations. Distributed measurements of the water height are inserted into the observer system through source terms of Luenberger type. Our main contribution is to show exponential convergence of the state of the observer system towards the original system state in the long time limit for a 2x2-system of nonlinear hyperbolic balance laws, i.e., we reconstruct the complete system state from measurements of one state variable. The proof is based on estimating the difference between the observer system and the original system via a suitable extension of the relative energy method.
Using energy-consistent coupling conditions and transforming the system to a Hamiltonian formulation, the synchronization result can be extended to star-shaped networks. This might have an application in flood modeling of river systems or control of irrigation systems.