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Sharp Observability Estimates for Heat Equations

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Abstract

The goal of this article is to derive new estimates for the cost of observability of heat equations. We have developed a new method allowing one to show that when the corresponding wave equation is observable, the heat equation is also observable. This method allows one to describe the explicit dependence of the observability constant on the geometry of the problem (the domain in which the heat process evolves and the observation subdomain). We show that our estimate is sharp in some cases, particularly in one space dimension and in the multi-dimensional radially symmetric case. Our result extends those in Fattorini and Russell (Arch Rational Mech Anal 43:272–292, 1971) to the multi-dimensional setting and improves those available in the literature, namely those by Miller (J Differ Equ 204(1):202–226, 2004; SIAM J Control Optim 45(2):762–772, 2006; Atti Accad Naz Lincei Cl Sci Fis Mat Natur Rend Lincei (9) Mat Appl 17(4):351–366, 2006) and Tenenbaum and Tucsnak (J Differ Equ 243(1):70–100, 2007). Our approach is based on an explicit representation formula of some solutions of the wave equation in terms of those of the heat equation, in contrast to the standard application of transmutation methods, which uses a reverse representation of the heat solution in terms of the wave one. We shall also explain how our approach applies and yields some new estimates on the cost of observability in the particular case of the unit square observed from one side. We will also comment on the applications of our techniques to controllability properties of heat-type equations.

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Authors and Affiliations

  1. CNRS, Institut de Mathématiques de Toulouse UMR 5219, 31062, Toulouse, France

    Sylvain Ervedoza

  2. Université de Toulouse, UPS, INSA, INP, ISAE, UT1, UTM, IMT, 31062, Toulouse, France

    Sylvain Ervedoza

  3. IKERBASQUE, Basque Foundation for Science, 48011, Bilbao, Basque Country, Spain

    Enrique Zuazua

  4. Basque Center for Applied Mathematics (BCAM), Bizkaia Technology Park, Building 500, 48160, Derio, Basque Country, Spain

    Enrique Zuazua

Authors
  1. Sylvain Ervedoza
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  2. Enrique Zuazua
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Correspondence to Sylvain Ervedoza.

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Communicated by C. Le Bris

This paper has mainly been developed while Sylvain Ervedoza was a visiting fellow at the Basque Center for Applied Mathematics (BCAM). He has been partially supported by the Agence Nationale de la Recherche (ANR, France), Project C-QUID number BLAN-3-139579 and Project CISIFSnumber NT09-437023. Enrique Zuazua was also partially supported by Grant MTM2008-03541 of the MICINN (Spain), project PI2010-04 of the Basque Government, the ERC Advanced Grant FP7-246775 NUMERIWAVES, and the ESF Research Networking Program OPTPDE.

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Ervedoza, S., Zuazua, E. Sharp Observability Estimates for Heat Equations. Arch Rational Mech Anal 202, 975–1017 (2011). https://doi.org/10.1007/s00205-011-0445-8

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