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Chinese Annals of Mathematics, Series B

, Volume 39, Issue 2, pp 281–296 | Cite as

Internal Controllability for Parabolic Systems Involving Analytic Non-local Terms

  • Pierre Lissy
  • Enrique Zuazua
Article
  • 106 Downloads

Abstract

This paper deals with the problem of internal controllability of a system of heat equations posed on a bounded domain with Dirichlet boundary conditions and perturbed with analytic non-local coupling terms. Each component of the system may be controlled in a different subdomain. Assuming that the unperturbed system is controllable—a property that has been recently characterized in terms of a Kalman-like rank condition—the authors give a necessary and sufficient condition for the controllability of the coupled system under the form of a unique continuation property for the corresponding elliptic eigenvalue system. The proof relies on a compactness-uniqueness argument, which is quite unusual in the context of parabolic systems, previously developed for scalar parabolic equations. The general result is illustrated by two simple examples.

Keywords

Parabolic systems Non-local potentials Analyticity Null controllability Kalman rank condition Spectral unique continuation 

2000 MR Subject Classification

35K40 93B05 93B07 

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References

  1. [1]
    Ammar-Khodja, F., Benabdallah, A., González-Burgos, M. and de Teresa, L., Recent results on the controllability of linear coupled parabolic problems: A survey, Mathematical Control and Related Fields, 1(3), 2011, 267–306.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Ammar-Khodja, F., Benabdallah, A., González-Burgos, M. and de Teresa, L., The Kalman condition for the boundary controllability of coupled parabolic systems, bounds on biorthogonal families to complex matrix exponentials, J. Math. Pures Appl. (9), 96(6), 2011, 555–590.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Ammar-Khodja, F., Benabdallah, A., González-Burgos, M. and de Teresa, L., Minimal time for the null controllability of parabolic systems: The effect of the condensation index of complex sequences, J. Funct. Anal., 267(7), 2014, 2077–2151.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Ammar Khodja, F., Benabdallah, A., González-Burgos, M. and de Teresa, L., New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence, J. Math. Anal. Appl., 444(2), 2016, 1071–1113.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Benabdallah, A., Boyer, F., González-Burgos, M. and Olive, G., Sharp estimates of the one-dimensional boundary control cost for parabolic systems and application to the N-dimensional boundary null controllability in cylindrical domains, SIAM J. Control Optim., 52(5), 2014, 2970–3001.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Boyer, F. and Olive, G., Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients, Math. Control Relat. Fields, 4(3), 2014, 263–287.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Ciarlet, P. G., Linear and Nonlinear Functional Analysis with Applications, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2013.Google Scholar
  8. [8]
    Coron, J.-M., Guerrero, S. and Rosier, L., Null controllability of a parabolic system with a cubic coupling term, SIAM Journal on Control and Optimization, 48(8), 2010, 5629–5653.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Coron, J.-M. and Guilleron, J.-P., Control of three heat equations coupled with two cubic nonlinearities, SIAM J. Control Optim., 55(2), 2016, 989–1019.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Duprez, M. and Lissy, P., Indirect controllability of some linear parabolic systems of m equations with m − 1 controls involving coupling terms of zero or first order, J. Math. Pures Appl. (9), 106(5), 2016, 905–934.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Duprez, M. and Lissy, P., Positive and negative results on the internal controllability of parabolic equations coupled by zero and first order terms, J. Evol. Equ., 2016, 1–22, DOI: 10.1007/s00028-017-0415-1.Google Scholar
  12. [12]
    Ervedoza, S. and Zuazua, E., Sharp observability estimates for heat equations, Archive for Rational Mechanics and Analysis, 202, 2011, 975–1017.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Fattorini, H. O., Some remarks on complete controllability, SIAM J. Control, 4(4), 1966, 686–694.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Fernández-Cara, E., González-Burgos, M. and de Teresa, L., Controllability of linear and semilinear nondiagonalizable parabolic systems, ESAIM Control Optim. Calc. Var., 21(4), 2015, 1178–1204.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Fernández-Cara, E., Lü, Q. and Zuazua, E., Null controllability of linear heat and wave equations with nonlocal spatial terms, SIAM J. Control Optim., 54(4), 2016, 2009–2019.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Fernández-Cara, E. and Zuazua, E., The cost of approximate controllability for heat equations: The linear case, Adv. Differential Equations, 5(4–6), 2000, 465–514.MathSciNetzbMATHGoogle Scholar
  17. [17]
    Ladyzenskaja, O. A., Solonnikov, V. A. and Ural’ceva, N. N., Linear and quasilinear equations of parabolic type, 23, American Mathematical Society, Providence, R I., 1968.CrossRefGoogle Scholar
  18. [18]
    Léautaud, M., Spectral inequalities for non-selfadjoint elliptic operators and application to the nullcontrollability of parabolic systems, J. Funct. Anal., 258(8), 2010, 2739–2778.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Lions, J.-L., Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30(1), 1988, 1–68.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Lorenzi, A., Two severely ill-posed linear parabolic problems, Alexandru Myller Mathematical Seminar, AIP Conf. Proc., 1329, Amer. Inst. Phys., Melville, NY, 2011, 150–169.Google Scholar
  21. [21]
    Lissy, P. and Zuazua, E., Internal observability for coupled systems of linear partial differential equations, HAL, 2017, https://hal.archives-ouvertes.fr/hal-01480301/document.Google Scholar
  22. [22]
    Micu, S. and Takahashi, T., Local controllability to stationary trajectories of a one-dimensional simplified model arising in turbulence, HAL, 2017, https://hal.archives-ouvertes.fr/hal-01572317.Google Scholar
  23. [23]
    Miller, L., A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups, Discrete Con- tin. Dyn. Syst. Ser. B, 14(4), 2010, 1465–1485.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Okubo, A. and Levin, S. A., Diffusion and Ecological Problems: Modern Perspectives, Interdisciplinary Applied Mathematics, 14, Springer-Verlag, New York, 2001.CrossRefGoogle Scholar
  25. [25]
    Russell, D. L., Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20(4), 1978, 639–739.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Zuazua, E., Stable observation of additive superpositions of partial differential equations, Systems Control Lett., 93, 2016, 21–29.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Fudan University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CeremadeUniversité Paris-Dauphine & CNRS UMR 7534, PSLParisFrance
  2. 2.DeustoTechUniversity of DeustoBilbao, Basque CountrySpain
  3. 3.Departamento de MatemáticasUniversidad Autónoma de MadridMadridSpain
  4. 4.Facultad IngenieríaUniversidad de DeustoBasque CountrySpain
  5. 5.UPMC Univ. Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis LionsSorbonne UniversitésParisFrance

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