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Controllability and observability of an artificial advection–diffusion problem

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Abstract

In this paper we study the controllability of an artificial advection–diffusion system through the boundary. Suitable Carleman estimates give us the observability of the adjoint system in the one dimensional case. We also study some basic properties of our problem such as backward uniqueness and we get an intuitive result on the control cost for vanishing viscosity.

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References

  1. Coron JM (2007) Control and nonlinearity. In: Mathematical Surveys and Monographs, vol 136. AMS, USA

  2. Coron JM, Guerrero S (2005) Singular optimal control: a linear 1-D parabolic–hyperbolic example. Asymptot Anal 44(3–4): 237–257

    MathSciNet  MATH  Google Scholar 

  3. Danchin R (1997) Poches de tourbillon visqueuses. J Math Pures Appl 76: 609–647

    MathSciNet  MATH  Google Scholar 

  4. Dolecki S, Russell D (1977) A general theory of observation and control. SIAM J Contr Optim 15(2): 185–220

    Article  MathSciNet  MATH  Google Scholar 

  5. Fursikov AV, Imanuvilov O (1996) Controllability of evolution equations. In: Lecture Notes Series, vol 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul

  6. Glass O (2010) A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit. J Funct Anal 258: 852–868

    Article  MathSciNet  MATH  Google Scholar 

  7. Glass O, Guerrero S (2007) On the uniform controllability of the Burgers equation. SIAM J Contr Optim 46(4): 1211–1238

    Article  MathSciNet  MATH  Google Scholar 

  8. Halpern L (1986) Artificial boundary for the linear advection diffusion equation. Math Comput 46(174): 425–438

    Article  MATH  Google Scholar 

  9. Halpern L, Schatzmann M (1989) Artificial boundary conditions for incompressible viscous flows. SIAM J Math Anal 20(2): 308–353

    Article  MathSciNet  MATH  Google Scholar 

  10. Lions JL (1988) Contrôlabilité exacte, stabilisation et perturbations de systèmes distribués. RMA vol 8. Masson, Paris

  11. Lions JL, Magenes E (1968) Problèmes aux limites non homogènes et applications, vol I, II (French). Dunod, Paris

  12. Lions JL, Malgrange B (1960) Sur l’unicité rétrograde dans les problèmes mixtes paraboliques (French). Math Scand 8: 277–286

    MathSciNet  Google Scholar 

  13. Miller L (2005) On the null-controllability of the heat equation in unbounded domains. Bull Sci Math 129(2): 175–185

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Lycée du parc des Loges, 1, boulevard des Champs-Élysées, 91012, Évry, France

    Pierre Cornilleau

  2. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 75252, Paris Cédex 05, France

    Sergio Guerrero

Authors
  1. Pierre Cornilleau
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  2. Sergio Guerrero
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Correspondence to Sergio Guerrero.

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Cornilleau, P., Guerrero, S. Controllability and observability of an artificial advection–diffusion problem. Math. Control Signals Syst. 24, 265–294 (2012). https://doi.org/10.1007/s00498-012-0076-0

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  • DOI: https://doi.org/10.1007/s00498-012-0076-0

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