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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Alessandrini G., Escauriaza L.: Null-controllability of one-dimensional parabolic equations. ESAIM Control Optim. Calc. Var. 14(2), 284–293 (2008)
Allibert B.: Contrôle analytique de l’équation des ondes sur des surfaces de révolution. C. R. Acad. Sci. Paris Sér. I Math. 322(9), 835–838 (1996)
Allibert B.: Analytic controllability of the wave equation over a cylinder. ESAIM Control Optim. Calc. Var. 4, 177–207 (1999)
Bardos C., Lebeau G., Rauch J.: Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30(5), 1024–1065 (1992)
Burq N., Burq N., Burq N.: Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes. C. R. Acad. Sci. Paris Sér. I Math. 325(7), 749–752 (1997)
Castro C., Zuazua E.: Concentration and lack of observability of waves in highly heterogeneous media. Arch. Rational Mech. Anal. 164(1), 39–72 (2002)
Chen X.-Y.: A strong unique continuation theorem for parabolic equations. Math. Ann. 311(4), 603–630 (1998)
Coron J.-M., Guerrero S.: Local null controllability of the two-dimensional Navier-Stokes system in the torus with a control force having a vanishing component. J. Math. Pures Appl. (9) 92(5), 528–545 (2009)
Dáger R., Zuazua E.: Wave propagation, observation and control in 1-d flexible multi-structures. Mathématiques & Applications, Vol. 50. Springer, Berlin (2006)
Ervedoza S., Zuazua E.: A systematic method for building smooth controls for smooth data. Discrete Contin. Dyn. Syst. Ser. B 14(4), 1375–1401 (2010)
Ervedoza, S., Zuazua, E.: Observability of heat processes by transmutation without geometric restrictions. Math. Control and Related Fields 1(2), (2011)
Fattorini H.O., Russell D.L.: Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Rational Mech. Anal. 43, 272–292 (1971)
Fernández-Cara E., Zuazua E.: The cost of approximate controllability for heat equations: the linear case. Adv. Differ. Equ. 5(4–6), 465–514 (2000)
Fernández-Cara E., Zuazua E.: On the null controllability of the one-dimensional heat equation with BV coefficients. Comput. Appl. Math. 21(1), 167–190 (2002) Special issue in memory of Jacques-Louis Lions
Fursikov A.V., Imanuvilov O.Y.: Controllability of evolution equations. Lecture Notes Series, Vol. 34. Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul (1996)
Haraux A.: A generalized internal control for the wave equation in a rectangle. J. Math. Anal. Appl. 153(1), 190–216 (1990)
John F.: Partial Differential Equations. Applied Mathematical Sciences, Vol. 1, 4th edn. Springer, New York (1982)
Frank Jones B. Jr.: A fundamental solution for the heat equation which is supported in a strip. J. Math. Anal. Appl. 60(2), 314–324 (1977)
Lang S.: Introduction to Diophantine Approximations. Addison-Wesley Publishing Co., Reading (1966)
Lebeau G.: Contrôle analytique. I. Estimations a priori. Duke Math. J. 68(1), 1–30 (1992)
Lebeau G.: Contrôle de l’équation de Schrödinger . J. Math. Pures Appl. (9) 71(3), 267–291 (1992)
Lebeau G., Robbiano L.: Contrôle exact de l’équation de la chaleur. Commun. Partial Differ. Equ. 20(1–2), 335–356 (1995)
Lions, J.-L.: Contrôlabilité exacte, Stabilisation et Perturbations de Systèmes Distribués. Tome 1. Contrôlabilité exacte, volume RMA 8. Masson, 1988
Micu S., de Teresa L.: A spectral study of the boundary controllability of the linear 2-D wave equation in a rectangle. Asymptot. Anal. 66(3–4), 139–160 (2010)
Micu S., Zuazua E.: On the controllability of a fractional order parabolic equation. SIAM J. Control Optim. 44(6), 1950–1972 (2006)
Micu S., Zuazua E.: Regularity issues for the null-controllability of the linear 1-d heat equation. Syst. Control Lett. 60, 406–413 (2011). doi:10.1016/j.sysconle.2011.03.005
Miller L.: A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups. Discrete Contin. Dyn. Syst. Ser. B 14(4), 1465–1485 (2010)
Miller L.: Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time. J. Differ. Equ. 204(1), 202–226 (2004)
Miller L.: On the null-controllability of the heat equation in unbounded domains. Bull. Sci. Math. 129(2), 175–185 (2005)
Miller L.: The control transmutation method and the cost of fast controls. SIAM J. Control Optim. 45(2), 762–772 (2006)
Miller L.: On exponential observability estimates for the heat semigroup with explicit rates. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl. (9) 17(4), 351–366 (2006)
Miller L.: On the controllability of anomalous diffusions generated by the fractional Laplacian. Math. Control Signals Syst. 18(3), 260–271 (2006)
Munch A., Zuazua E.: Numerical approximation of null controls for the heat equation through transmutation. J. Inverse Probl. 26(8), 085018 (2010). doi:10.1088/0266-5611/26/8/085018
Phung, K.D.: Waves, damped wave and observation. In: Li, T.-T., Peng, Y.-J., Rao, B.-P. (Eds.) Some Problems on Nonlinear Hyperbolic Equations and Applications. Series in Contemporary Applied Mathematics CAM 15 (2010)
Rauch J., Zhang X., Zhang X., Zhang X.: Polynomial decay for a hyperbolic-parabolic coupled system. J. Math. Pures Appl. (9) 84(4), 407–470 (2005)
Robbiano L.: Fonction de coût et contrôle des solutions des équations hyperboliques. Asympt. Anal. 10(2), 95–115 (1995)
Russell D.L.: A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Stud. Appl. Math. 52, 189–211 (1973)
Seidman T.I.: Time-invariance of the reachable set for linear control problems. J. Math. Anal. Appl. 72(1), 17–20 (1979)
Tenenbaum G., Tucsnak M.: New blow-up rates for fast controls of Schrödinger and heat equations. J. Differ. Equ. 243(1), 70–100 (2007)
Widder D.V.: The role of the Appell transformation in the theory of heat conduction. Trans. Am. Math. Soc. 109, 121–134 (1963)
Zuazua E.: Exact controllability for semilinear wave equations in one space dimension. Ann. Inst. H. Poincaré Anal. Non Linéaire 10(1), 109–129 (1993)
Zuazua, E.: Some results and open problems on the controllability of linear and semilinear heat equations. In: Carleman Estimates and Applications to Uniqueness and Control Theory (Cortona, 1999). Progr. Nonlinear Differential Equations Appl.,Vol. 46, pp. 191–211. Birkhäuser Boston, Boston, 2001
Zuazua, E.: Controllability and observability of partial differential equations: some results and open problems. In: Dafermos, C.M., Elsevier Science Feireisl, E. (Eds.) Handbook of Differential Equations, Vol. 3, pp. 527–621 (2006)
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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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DOI: https://doi.org/10.1007/s00205-011-0445-8