December  2019, 9(4): 673-696. doi: 10.3934/mcrf.2019046

Backward uniqueness results for some parabolic equations in an infinite rod

Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, UPS IMT F-31062 Toulouse Cedex 9, France

* Corresponding author: Sylvain Ervedoza

Received  June 2018 Revised  August 2019 Published  November 2019

Fund Project: The first author is partially supported by IFSMACS ANR-15-CE40-0010 of the French National Research Agency (ANR) and both authors are supported by the CIMI Labex, Toulouse, France, under grant ANR-11-LABX-0040-CIMI

The goal of this article is to provide backward uniqueness results for several models of parabolic equations set on the half line, namely the heat equation, and the heat equation with quadratic potential and with purely imaginary quadratic potentials, with non-homogeneous boundary conditions. Such result can thus also be interpreted as a strong lack of controllability on the half line, as it shows that only the trivial initial datum can be steered to zero. Our results are based on the explicit knowledge of the kernel of each equation, and standard arguments from complex analysis, namely the Phragmén-Lindelöf principle.

Citation: Jérémi Dardé, Sylvain Ervedoza. Backward uniqueness results for some parabolic equations in an infinite rod. Mathematical Control & Related Fields, 2019, 9 (4) : 673-696. doi: 10.3934/mcrf.2019046
References:
[1]

H. AikawaN. Hayashi and S. Saitoh, The Bergman space on a sector and the heat equation, Complex Variables Theory Appl., 15 (1990), 27-36.  doi: 10.1080/17476939008814430.  Google Scholar

[2]

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K. BeauchardP. Cannarsa and R. Guglielmi, Null controllability of Grushin-type operators in dimension two, J. Eur. Math. Soc. (JEMS), 16 (2014), 67-101.  doi: 10.4171/JEMS/428.  Google Scholar

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K. Beauchard, J. Dardé and S. Ervedoza, Minimal time issues for the observability of Grushin-type equations, December 2017. Google Scholar

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K. BeauchardB. HelfferR. Henry and L. Robbiano, Degenerate parabolic operators of Kolmogorov type with a geometric control condition, ESAIM Control Optim. Calc. Var., 21 (2015), 487-512.  doi: 10.1051/cocv/2014035.  Google Scholar

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K. BeauchardL. Miller and M. Morancey, 2D Grushin-type equations: Minimal time and null controllable data, J. Differential Equations, 259 (2015), 5813-5845.  doi: 10.1016/j.jde.2015.07.007.  Google Scholar

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K. Beauchard and K. Pravda-Starov, Null-controllability of non-autonomous Ornstein-Uhlenbeck equations, J. Math. Anal. Appl., 456 (2017), 496–524. doi: 10.1016/j.jmaa.2017.07.014.  Google Scholar

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T. Duyckaerts and L. Miller, Resolvent conditions for the control of parabolic equations, J. Funct. Anal., 263 (2012), 3641-3673.  doi: 10.1016/j.jfa.2012.09.003.  Google Scholar

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Ju. V. Egorov, Some problems in the theory of optimal control, Ž. Vyčisl. Mat. i Mat. Fiz., 3 (1963), 887-904.   Google Scholar

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L. EscauriazaC. E. KenigG. Ponce and L. Vega, Hardy's uncertainty principle, convexity and Schrödinger evolutions, J. Eur. Math. Soc. (JEMS), 10 (2008), 883-907.  doi: 10.4171/JEMS/134.  Google Scholar

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L. EscauriazaG. Seregin and V. Šverák, Backward uniqueness for the heat operator in half-space, Algebra i Analiz, 15 (2003), 201-214.   Google Scholar

[17]

H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal., 43 (1971), 272-292.  doi: 10.1007/BF00250466.  Google Scholar

[18]

E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Adv. Differential Equations, 5 (2000), 465-514.   Google Scholar

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A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, volume 34 of Lecture Notes Series, Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul, 1996.  Google Scholar

[20]

B. Helffer, Semi-classical Analysis for the Schrödinger Operator and Applications, volume 1336 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1988. doi: 10.1007/BFb0078115.  Google Scholar

[21]

L. IskauriazaG. A. Serëgin and V. Shverak, $L_{3, \infty}$-solutions of Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk, 58 (2003), 3-44.  doi: 10.1070/RM2003v058n02ABEH000609.  Google Scholar

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F. John, Partial Differential Equations, volume 1 of Applied Mathematical Sciences, Springer-Verlag, New York, fourth edition, 1982. doi: 10.1007/978-1-4684-9333-7.  Google Scholar

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B. Frank Jones and Jr ., A fundamental solution for the heat equation which is supported in a strip, J. Math. Anal. Appl., 60 (1977), 314-324.  doi: 10.1016/0022-247X(77)90021-X.  Google Scholar

[24]

A. Koenig, Non null controllability of the Grushin equation in 2D, C. R. Math. Acad. Sci. Paris, 355 (2017), 1215–1235. https://arXiv.org/abs/1701.06467, 2017. doi: 10.1016/j.crma.2017.10.021.  Google Scholar

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C. Laurent and M. Leautaud, Tunneling estimates and approximate controllability for hypoelliptic equations, https://arXiv.org/abs/1703.10797, March 2017. Google Scholar

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J. Le Rousseau and I. Moyano, Null-controllability of the Kolmogorov equation in the whole phase space, J. Differential Equations, 260 (2016), 3193-3233.  doi: 10.1016/j.jde.2015.09.062.  Google Scholar

[27]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356.  doi: 10.1080/03605309508821097.  Google Scholar

[28]

B. Ya. Levin, Lectures on Entire Functions, volume 150 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1996. In collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko, Translated from the Russian manuscript by Tkachenko.  Google Scholar

[29]

L. Li and V. Šverák, Backward uniqueness for the heat equation in cones, Comm. Partial Differential Equations, 37 (2012), 1414-1429.  doi: 10.1080/03605302.2011.635323.  Google Scholar

[30]

S. Micu and E. Zuazua, On the lack of null-controllability of the heat equation on the half-line, Trans. Amer. Math. Soc., 353 (2001), 1635–1659 (electronic). doi: 10.1090/S0002-9947-00-02665-9.  Google Scholar

[31]

S. Micu and E. Zuazua, On the controllability of a fractional order parabolic equation, SIAM J. Control Optim., 44 (2006), 1950–1972 (electronic). doi: 10.1137/S036301290444263X.  Google Scholar

[32]

L. Miller, On the null-controllability of the heat equation in unbounded domains, Bull. Sci. Math., 129 (2005), 175-185.  doi: 10.1016/j.bulsci.2004.04.003.  Google Scholar

[33]

L. Miller, Unique continuation estimates for the Laplacian and the heat equation on non-compact manifolds, Math. Res. Lett., 12 (2005), 37-47.  doi: 10.4310/MRL.2005.v12.n1.a4.  Google Scholar

[34]

L. Miller, On the controllability of anomalous diffusions generated by the fractional Laplacian, Math. Control Signals Systems, 18 (2006), 260-271.  doi: 10.1007/s00498-006-0003-3.  Google Scholar

[35]

L. Miller, Unique continuation estimates for sums of semiclassical eigenfunctions and null-controllability from cones, 20 pages, 1 figure, AMS-LaTeX., November 2008. Google Scholar

[36]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[37]

J. Rauch, Partial Differential Equations, volume 128 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0953-9.  Google Scholar

[38]

A. Rüland, On the backward uniqueness property for the heat equation in two-dimensional conical domains, Manuscripta Math., 147 (2015), 415-436.  doi: 10.1007/s00229-015-0764-4.  Google Scholar

[39]

G. Seregin and V. Šverák, The Navier-Stokes equations and backward uniqueness, In Nonlinear Problems in Mathematical Physics and Related Topics, II, volume 2 of Int. Math. Ser. (N. Y.), pages 353–366. Kluwer/Plenum, New York, 2002.  Google Scholar

[40]

G. Wang, M. Wang, C. Zhang and Y. Zhang, Observable set, observability, interpolation inequality and spectral inequality for the heat equation in $\mathbb{R}^n$, J. Math. Pures Appl., 126 (2019), 144–194, https://arXiv.org/abs/1711.04279. doi: 10.1016/j.matpur.2019.04.009.  Google Scholar

[41]

D. V. Widder, The role of the Appell transformation in the theory of heat conduction, Trans. Amer. Math. Soc., 109 (1963), 121-134.  doi: 10.1090/S0002-9947-1963-0154068-2.  Google Scholar

[42]

J. Wu and W. Wang, On backward uniqueness for the heat operator in cones, J. Differential Equations, 258 (2015), 224-241.  doi: 10.1016/j.jde.2014.09.011.  Google Scholar

show all references

References:
[1]

H. AikawaN. Hayashi and S. Saitoh, The Bergman space on a sector and the heat equation, Complex Variables Theory Appl., 15 (1990), 27-36.  doi: 10.1080/17476939008814430.  Google Scholar

[2]

K. Beauchard, Null controllability of Kolmogorov-type equations, Math. Control Signals Systems, 26 (2014), 145-176.  doi: 10.1007/s00498-013-0110-x.  Google Scholar

[3]

K. BeauchardP. Cannarsa and R. Guglielmi, Null controllability of Grushin-type operators in dimension two, J. Eur. Math. Soc. (JEMS), 16 (2014), 67-101.  doi: 10.4171/JEMS/428.  Google Scholar

[4]

K. Beauchard, J. Dardé and S. Ervedoza, Minimal time issues for the observability of Grushin-type equations, December 2017. Google Scholar

[5]

K. BeauchardB. HelfferR. Henry and L. Robbiano, Degenerate parabolic operators of Kolmogorov type with a geometric control condition, ESAIM Control Optim. Calc. Var., 21 (2015), 487-512.  doi: 10.1051/cocv/2014035.  Google Scholar

[6]

K. BeauchardL. Miller and M. Morancey, 2D Grushin-type equations: Minimal time and null controllable data, J. Differential Equations, 259 (2015), 5813-5845.  doi: 10.1016/j.jde.2015.07.007.  Google Scholar

[7]

K. Beauchard and K. Pravda-Starov, Null-controllability of non-autonomous Ornstein-Uhlenbeck equations, J. Math. Anal. Appl., 456 (2017), 496–524. doi: 10.1016/j.jmaa.2017.07.014.  Google Scholar

[8]

L. S. Boulton, Non-self-adjoint harmonic oscillator, compact semigroups and pseudospectra, J. Operator Theory, 47 (2002), 413-429.   Google Scholar

[9] E. B. Davies, Heat Kernels and Spectral Theory, volume 92 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1989.  doi: 10.1017/CBO9780511566158.  Google Scholar
[10]

E. B. Davies, Pseudo-spectra, the harmonic oscillator and complex resonances, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 585-599.  doi: 10.1098/rspa.1999.0325.  Google Scholar

[11]

E. B. Davies, Semi-classical states for non-self-adjoint Schrödinger operators, Comm. Math. Phys., 200 (1999), 35-41.  doi: 10.1007/s002200050521.  Google Scholar

[12]

E. B. Davies and A. B. J. Kuijlaars, Spectral asymptotics of the non-self-adjoint harmonic oscillator, J. London Math. Soc. (2), 70 (2004), 420-426.  doi: 10.1112/S0024610704005381.  Google Scholar

[13]

T. Duyckaerts and L. Miller, Resolvent conditions for the control of parabolic equations, J. Funct. Anal., 263 (2012), 3641-3673.  doi: 10.1016/j.jfa.2012.09.003.  Google Scholar

[14]

Ju. V. Egorov, Some problems in the theory of optimal control, Ž. Vyčisl. Mat. i Mat. Fiz., 3 (1963), 887-904.   Google Scholar

[15]

L. EscauriazaC. E. KenigG. Ponce and L. Vega, Hardy's uncertainty principle, convexity and Schrödinger evolutions, J. Eur. Math. Soc. (JEMS), 10 (2008), 883-907.  doi: 10.4171/JEMS/134.  Google Scholar

[16]

L. EscauriazaG. Seregin and V. Šverák, Backward uniqueness for the heat operator in half-space, Algebra i Analiz, 15 (2003), 201-214.   Google Scholar

[17]

H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal., 43 (1971), 272-292.  doi: 10.1007/BF00250466.  Google Scholar

[18]

E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Adv. Differential Equations, 5 (2000), 465-514.   Google Scholar

[19]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, volume 34 of Lecture Notes Series, Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul, 1996.  Google Scholar

[20]

B. Helffer, Semi-classical Analysis for the Schrödinger Operator and Applications, volume 1336 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1988. doi: 10.1007/BFb0078115.  Google Scholar

[21]

L. IskauriazaG. A. Serëgin and V. Shverak, $L_{3, \infty}$-solutions of Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk, 58 (2003), 3-44.  doi: 10.1070/RM2003v058n02ABEH000609.  Google Scholar

[22]

F. John, Partial Differential Equations, volume 1 of Applied Mathematical Sciences, Springer-Verlag, New York, fourth edition, 1982. doi: 10.1007/978-1-4684-9333-7.  Google Scholar

[23]

B. Frank Jones and Jr ., A fundamental solution for the heat equation which is supported in a strip, J. Math. Anal. Appl., 60 (1977), 314-324.  doi: 10.1016/0022-247X(77)90021-X.  Google Scholar

[24]

A. Koenig, Non null controllability of the Grushin equation in 2D, C. R. Math. Acad. Sci. Paris, 355 (2017), 1215–1235. https://arXiv.org/abs/1701.06467, 2017. doi: 10.1016/j.crma.2017.10.021.  Google Scholar

[25]

C. Laurent and M. Leautaud, Tunneling estimates and approximate controllability for hypoelliptic equations, https://arXiv.org/abs/1703.10797, March 2017. Google Scholar

[26]

J. Le Rousseau and I. Moyano, Null-controllability of the Kolmogorov equation in the whole phase space, J. Differential Equations, 260 (2016), 3193-3233.  doi: 10.1016/j.jde.2015.09.062.  Google Scholar

[27]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356.  doi: 10.1080/03605309508821097.  Google Scholar

[28]

B. Ya. Levin, Lectures on Entire Functions, volume 150 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1996. In collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko, Translated from the Russian manuscript by Tkachenko.  Google Scholar

[29]

L. Li and V. Šverák, Backward uniqueness for the heat equation in cones, Comm. Partial Differential Equations, 37 (2012), 1414-1429.  doi: 10.1080/03605302.2011.635323.  Google Scholar

[30]

S. Micu and E. Zuazua, On the lack of null-controllability of the heat equation on the half-line, Trans. Amer. Math. Soc., 353 (2001), 1635–1659 (electronic). doi: 10.1090/S0002-9947-00-02665-9.  Google Scholar

[31]

S. Micu and E. Zuazua, On the controllability of a fractional order parabolic equation, SIAM J. Control Optim., 44 (2006), 1950–1972 (electronic). doi: 10.1137/S036301290444263X.  Google Scholar

[32]

L. Miller, On the null-controllability of the heat equation in unbounded domains, Bull. Sci. Math., 129 (2005), 175-185.  doi: 10.1016/j.bulsci.2004.04.003.  Google Scholar

[33]

L. Miller, Unique continuation estimates for the Laplacian and the heat equation on non-compact manifolds, Math. Res. Lett., 12 (2005), 37-47.  doi: 10.4310/MRL.2005.v12.n1.a4.  Google Scholar

[34]

L. Miller, On the controllability of anomalous diffusions generated by the fractional Laplacian, Math. Control Signals Systems, 18 (2006), 260-271.  doi: 10.1007/s00498-006-0003-3.  Google Scholar

[35]

L. Miller, Unique continuation estimates for sums of semiclassical eigenfunctions and null-controllability from cones, 20 pages, 1 figure, AMS-LaTeX., November 2008. Google Scholar

[36]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[37]

J. Rauch, Partial Differential Equations, volume 128 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0953-9.  Google Scholar

[38]

A. Rüland, On the backward uniqueness property for the heat equation in two-dimensional conical domains, Manuscripta Math., 147 (2015), 415-436.  doi: 10.1007/s00229-015-0764-4.  Google Scholar

[39]

G. Seregin and V. Šverák, The Navier-Stokes equations and backward uniqueness, In Nonlinear Problems in Mathematical Physics and Related Topics, II, volume 2 of Int. Math. Ser. (N. Y.), pages 353–366. Kluwer/Plenum, New York, 2002.  Google Scholar

[40]

G. Wang, M. Wang, C. Zhang and Y. Zhang, Observable set, observability, interpolation inequality and spectral inequality for the heat equation in $\mathbb{R}^n$, J. Math. Pures Appl., 126 (2019), 144–194, https://arXiv.org/abs/1711.04279. doi: 10.1016/j.matpur.2019.04.009.  Google Scholar

[41]

D. V. Widder, The role of the Appell transformation in the theory of heat conduction, Trans. Amer. Math. Soc., 109 (1963), 121-134.  doi: 10.1090/S0002-9947-1963-0154068-2.  Google Scholar

[42]

J. Wu and W. Wang, On backward uniqueness for the heat operator in cones, J. Differential Equations, 258 (2015), 224-241.  doi: 10.1016/j.jde.2014.09.011.  Google Scholar

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