  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. 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.
