Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the lack of null-controllability of the heat equation on the half-line


Authors: Sorin Micu and Enrique Zuazua
Journal: Trans. Amer. Math. Soc. 353 (2001), 1635-1659
MSC (2000): Primary 35B37, 35K05
DOI: https://doi.org/10.1090/S0002-9947-00-02665-9
Published electronically: November 21, 2000
MathSciNet review: 1806726
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

We consider the linear heat equation on the half-line with a Dirichlet boundary control. We analyze the null-controllability problem. More precisely, we study the class of initial data that may be driven to zero in finite time by means of an appropriate choice of the $L^2$ boundary control. We rewrite the system on the similarity variables that are a common tool when analyzing asymptotic problems. Next, the control problem is reduced to a moment problem which turns out to be critical since it concerns the family of real exponentials $\{e^{jt}\}_{j\geq1}$ in which the usual summability condition on the inverses of the eigenvalues does not hold. Roughly speaking, we prove that controllable data have Fourier coefficients that grow exponentially for large frequencies. This result is in contrast with the existing ones for bounded domains that guarantee that every initial datum belonging to a Sobolev space of negative order may be driven to zero in an arbitrarily small time.


References [Enhancements On Off] (What's this?)

  • 1. S.A. Avdonin and S.A. Ivanov, Families of exponentials. The method of moments in controllability problems for distributed parameter systems, Cambridge University Press, 1995. MR 97b:93002
  • 2. R.P. Boas, Entire functions, Academic Press, New York, 1954. MR 16:914f
  • 3. C. Bandle and H.A. Levine, On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains, Trans. Amer. Math. Soc., 316 (2) (1989), 595-622. MR 90c:35118
  • 4. M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of the heat equation, Nonlinear Analysis TMA, 11 (10) (1987), 1103-1133. MR 90a:3512q
  • 5. H. Fattorini and D.L. Russell, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations, Quarterly J. Appl. Math., 32 (1974), 45-69. MR 58:23325
  • 6. E. Fernández-Cara, Null controllability of the semilinear heat equation, ESAIM:COCV, 2 (1997), 87-107, (http://www.emath.fr/cocv/). MR 98d:93011
  • 7. E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Advances Diff. Eqs, 5 (2000), 465-514.
  • 8. E. Fernández-Cara and E. Zuazua, Null controllability of weakly blowing-up semilinear heat equations, preprint, 1999.
  • 9. A. Fursikov and O. Yu. Imanuvilov, Controllability of evolution equations, Lecture Notes Series #34, Research Institute of Mathematics, Global Analysis Research Center, Seoul National University, 1996. MR 97g:93002
  • 10. J.M. Ghidaglia, Some backward uniqueness results, Nonlinear Anal. TMA, 10 (1986), 777-790. MR 87m:34083
  • 11. E. N. Güichal, A lower bound on the norm of the control operator for the heat equation, J. Math. Anal. Appl., 110 (1985), 519-527. MR 87i:93069
  • 12. O. Yu. Imanuvilov and M. Yamamoto, On Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Preprint n $^{\underline{0}}$ 98-46, University of Tokyo, Graduate School of Mathematics, Japan, 1998.
  • 13. F. John, Partial Differential Equations, Springer Verlag, 1978. MR 87g:35002 (later ed.)
  • 14. B.F. Jones, Jr., A fundamental solution of the heat equation which is supported in a strip, J. Math. Anal. Appl., 60 (1977), 314-324. MR 56:9070
  • 15. O. Kavian, Remarks on the large time behavior of a nonlinear diffusion equation, Ann. Inst. Henri Poincaré, Analyse non linéaire, 4 (5) (1987), 432-452. MR 89b:35013
  • 16. W. Littman, Boundary control theory for hyperbolic and parabolic partial differential equations with constant coefficients, Annali Scuola Norm. Sup. Pisa, Serie IV, 3 (1978), 567-580. MR 80a:35023
  • 17. J.L. Lions and B. Malgrange, Sur l'unicité rétrograde dans les problèmes mixtes paraboliques, Math. Scan., 8 (1960), 277-286. MR 25:4269
  • 18. G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. P.D.E., 20 (1995), 335-356. MR 95m:93045
  • 19. S. Micu and E. Zuazua, On the lack of null-controllability of the heat equation on the half space, Portugal. Math., to appear.
  • 20. F.W.J. Olver, Asymptotics and special functions, Academic Press, 1974. MR 55:8655
  • 21. L. Rosier, Exact controllability for the linear Korteweg-de Vries equation on the half-line, SIAM J. Cont. Optim., 39 (2000), 331-351.
  • 22. D.L. Russell, Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions, SIAM Rev., 20 (1978), 639-739. MR 80c:93032
  • 23. L. Schwartz, Etude des sommes d'exponentielles, Hermann, Paris, 1959. MR 21:5116
  • 24. T.I. Seidman, S.A. Avdonin and S.A. Ivanov, The ``window problem'' for series of complex exponentials, preprint, 1998.
  • 25. L. de Teresa, Approximate controllability of a semilinear heat equation in $\mathbb{R}^N$, SIAM J. Cont. Optim., 36 (6) (1996), 855-884. MR 99j:93011
  • 26. L. de Teresa and E. Zuazua, Approximate controllability of the heat equation in unbounded domains, Nonlinear Anal. TMA, 37 (1999), 1059-1090. CMP 99:14
  • 27. K. Yosida, Functional Analisis, Springer Verlag, 1980. MR 82i:46002
  • 28. R.M. Young, An introduction to nonharmonic Fourier series, Academic Press, 1980. MR 81m:42027

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35B37, 35K05

Retrieve articles in all journals with MSC (2000): 35B37, 35K05


Additional Information

Sorin Micu
Affiliation: Departamento de Matemática Aplicada, Universidad Complutense, 28040 Madrid, Spain
Email: sorin@sunma4.mat.ucm.es

Enrique Zuazua
Affiliation: Departamento de Matemática Aplicada, Universidad Complutense, 28040 Madrid, Spain
Email: zuazua@eucmax.sim.ucm.es

DOI: https://doi.org/10.1090/S0002-9947-00-02665-9
Keywords: Heat equation, similarity variables, control, moments
Received by editor(s): July 21, 1999
Received by editor(s) in revised form: October 22, 1999
Published electronically: November 21, 2000
Additional Notes: The first author was supported by grants PB96-0663 and 303/1999 of CNCSU (Romania)
The second author was supported by grant PB96-0663 of the DGES (Spain)
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society