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We present a new continuous data assimilation algorithm based on ideas thathave been developed for designing finite-dimensional feedback controls fordissipative dynamical systems, in particular, in the context of theincompressible two-dimensional Navier--Stokes equations. These ideas aremotivated by the fact that dissipative dynamical systems possess finite numbersof determining parameters (degrees of freedom) such as modes, nodes and localspatial averages which govern their long-term behavior. Therefore, ouralgorithm allows the use of any type of measurement data for which a generaltype of approximation interpolation operator exists. Our main result providesconditions, on the finite-dimensional spatial resolution of the collected data,sufficient to guarantee that the approximating solution, obtained by ouralgorithm from the measurement data, converges to the unknown referencesolution over time. Our algorithm is also applicable in the context of signalsynchronization in which one can recover, asymptotically in time, the solution(signal) of the underlying dissipative system that is corresponding to acontinuously transmitted partial data.

Continuous Data Assimilation Using General Interpolant Observables