Continuous Data Assimilation Using General Interpolant Observables

We present a new continuous data assimilation algorithm based on ideas that have been developed for designing finite-dimensional feedback controls for dissipative dynamical systems, in particular, in the context of the incompressible two-dimensional Navier-Stokes equations. These ideas are motivated by the fact that dissipative dynamical systems possess finite numbers of determining parameters (degrees of freedom) such as modes, nodes and local spatial averages which govern their long-term behavior. Therefore, our algorithm allows the use of any type of measurement data for which a general type of approximation interpolation operator exists. Under the assumption that the observational measurements are free of noise, our main result provides conditions, on the finite-dimensional spatial resolution of the collected data, sufficient to guarantee that the approximating solution, obtained by our algorithm from the measurement data, converges to the unknown reference solution over time. Our algorithm is also applicable in the context of signal synchronization in which one can recover, asymptotically in time, the solution (signal) of the underlying dissipative system that is corresponding to a continuously transmitted partial data.