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Journal of Nonlinear Science

, Volume 24, Issue 2, pp 277–304 | Cite as

Continuous Data Assimilation Using General Interpolant Observables

  • Abderrahim Azouani
  • Eric OlsonEmail author
  • Edriss S. Titi
Article

Abstract

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.

Keywords

Determining modes Volume elements and nodes Continuous data assimilation Two-dimensional Navier–Stokes equations Signal synchronization 

Mathematics Subject Classification

35Q30 93C20 37C50 76B75 34D06 

Notes

Acknowledgements

The work of A.A. is supported in part by the DFG grants SFB-910 and SFB-947. E.S.T. is thankful to the kind hospitality of the Freie Universität Berlin, where this work was initiated. E.S.T. also acknowledges the partial support of the Alexander von Humboldt Stiftung Foundation, the Minerva Stiftung Foundation, and the National Science Foundation grants DMS-1009950, DMS-1109640 and DMS-1109645. We would also like to thank the anonymous referees for their careful reading and constructive comments.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Abderrahim Azouani
    • 1
  • Eric Olson
    • 2
    Email author
  • Edriss S. Titi
    • 3
    • 4
  1. 1.Freie Universität BerlinInstitute für Mathematik IBerlinGermany
  2. 2.Department of Mathematics and StatisticsUniversity of NevadaRenoUSA
  3. 3.Department of Mathematics and Department of Mechanical and Aerospace EngineeringUniversity of CaliforniaIrvineUSA
  4. 4.The Department of Computer Science and Applied MathematicsWeizmann Institute of ScienceRehovotIsrael

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