Exact Controllability of Parabolic Equations

  • Viorel Barbu
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 90)


This chapter is concerned with the presentation of some basic results on the exact internal and boundary controllability of linear and semilinear parabolic equations on smooth domains of \({\mathbb {R}}^d.\) The exact controllability of linear stochastic parabolic equations with linear multiplicative Gaussian noise is also briefly studied. The main ingredient to exact controllability is the observability inequality for the dual parabolic equations established in Chapter  2.


  1. 3.
    Ammar Khodja, F., Benabdallah, A., Dupaix, C., Kostin, L.: Controllability to the trajectories of two-phase field models by one control force. SIAMJ. Control Optim. 42(5), 1661–1680 (2003)Google Scholar
  2. 7.
    Aniţa, S., Barbu, V.: Null controllability of nonlinear convective heat equations. ESAIM: Control Optim. Calc. Var. 5, 157–173 (2000)MathSciNetCrossRefGoogle Scholar
  3. 8.
    Aniţa, S., Barbu, V.: Local exact controllability of a reaction diffusion system. Differ. Integr. Equ. 14(5), 577–587 (2001)Google Scholar
  4. 9.
    Aniţa, S., Tataru, D.: Null controllability of dissipative semilinear heat equation. Appl. Math. Optim. 46(2), 97–105 (2002)Google Scholar
  5. 10.
    Araruna, F.D., Boldrini, J.L., Calsavara, B.M.R.: Optimal control and controllability of a phase field system with one control force. Appl. Math. Optim. 70, 539–563 (2014)MathSciNetCrossRefGoogle Scholar
  6. 18.
    Barbu, V.: Analysis and Control of Nonlinear Infinite Dimensional Systems. Academic, Boston (1993)Google Scholar
  7. 20.
    Barbu, V.: Exact controllability of the superlinear heat equations. Appl. Math. Optim. 42, 73–89 (2000)MathSciNetCrossRefGoogle Scholar
  8. 22.
    Barbu, V.: Controllability of parabolic and Navier–Stokes equations. Sci. Math. Japonicae 56, 143–210 (2002)Google Scholar
  9. 23.
    Barbu, V.: Local controllability of the phase field system. Nonlinear Anal. Theory Methods Appl. 50, 363–372 (2002)MathSciNetCrossRefGoogle Scholar
  10. 26.
    Barbu, V.: Nonlinear Differential Equations of Monotone Type in Banach Spaces. Springer, New York (2010)CrossRefGoogle Scholar
  11. 31.
    Barbu, V.: Exact null internal controllability for the heat equation on unbounded convex domains. ESAIM: Control Optim. Calc. Var. 6(1), 222–235 (2013)MathSciNetCrossRefGoogle Scholar
  12. 34.
    Barbu, V., Iannelli, M.: Controllability of the heat equation with memory. Differ. Integr. Equ. 13, 1393–1412 (2000)Google Scholar
  13. 38.
    Barbu, V., Da Prato, G.: The generator of the transition semigroup corresponding to a stochastic partial differential equation. Commun. Part. Differ. Equ. 33, 1318–1330 (2008)Google Scholar
  14. 42.
    Barbu, V., Tubaro, L.: Exact controllability of stochastic differential equations with multiplicative noise (submitted)Google Scholar
  15. 43.
    Barbu, V., Rascanu, A., Tessitore, G.: Carleman estimates and controllability of stochastic heat equation with multiplicative noise. Appl. Math. Optim. 47, 97–120 (2003)Google Scholar
  16. 50.
    Beceanu, M.: Local exact controllability of nonlinear diffusion equation in 1-D. Abstr. Appl. Anal. 14, 793–811 (2003)Google Scholar
  17. 57.
    Caginalp, C.: An analysis of a phase field model of a free boundary. Arch. Ration. Mech. Anal. 92, 205–243 (1986)Google Scholar
  18. 58.
    Cannarsa, P., Fragnelli, G., Rocchetti, D.: Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form. J. Evol. Equ. 8, 583–616 (2008)MathSciNetCrossRefGoogle Scholar
  19. 59.
    Cannarsa, P., Martinez, P., Vancostenoble, J.: Global Carleman estimates for degenerate parabolic operators with applications. Mem. Am. Math. Soc. 239 (2016)MathSciNetCrossRefGoogle Scholar
  20. 60.
    Coron, J.M.: Control and Nonlinearity. Mathematical Survey and Monographs, vol. 136. American Mathematical Society, Providence (2007)Google Scholar
  21. 61.
    Coron, J.M., Guilleron, J.Ph.: Control of three heat equations coupled with two cubic nonlinearities. SIAM J. Control Optim. 55, 989–1019 (2017)MathSciNetCrossRefGoogle Scholar
  22. 62.
    Coron, J.M., Guerrero, S., Rossier, L.: Null controllability of a parabolic system with a cubic coupling term. SIAM J. Control Optim. 48, 5629–5635 (2010)MathSciNetCrossRefGoogle Scholar
  23. 63.
    Coron, J.-M., Díaz, J.I., Drici, A., Mingazzinbi, T.: Global null controllability of the 1-dimensional nonlinear slow diffusion equation. Chin. Ann. Math. B 34(3), 333–344 (2013)MathSciNetCrossRefGoogle Scholar
  24. 64.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimension. Cambridge University Press, Cambridge (2013)Google Scholar
  25. 65.
    Deimling, K.: Nonlinear Analysis. Springer, Berlin (1990)Google Scholar
  26. 66.
    Diaz, J.I., Henry, J., Ramos, A.M.: On the approximate controllability of some semilinear parabolic boundary-value problems. Appl. Math. Optim. 37, 71–97 (1998)MathSciNetCrossRefGoogle Scholar
  27. 67.
    Dubova, A., Osses, A., Puel, J.P.: Exact controllability to trajectories for semilinear heat equations with discontinuous coefficients. ESAIM: Control Optim. Calc. Var. 8, 621–661 (2002)Google Scholar
  28. 68.
    Dubova, A., Fernandez-Cara, E., Gonzales-Burgos, M., Zuazua, E.: On the controllability of parabolic systems with a nonlinear term involving the state and the gradient. SIAM J. Control Optim. 41, 718–819 (2002)Google Scholar
  29. 69.
    Fattorini, H.D., Russell, D.L.: Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Ration. Mech. Anal. 43, 272–292 (1971)Google Scholar
  30. 70.
    Fernandez, L.A., Zuazua, E., Approximate controllability for the semilinear heat equation involving gradient terms. J. Optim. Theory Appl. 101, 307–328 (1999)MathSciNetCrossRefGoogle Scholar
  31. 71.
    Fernandez-Cara, E.: Null controllability of the semilinear heat equation. ESAIM: Control Optim. Calc. Var. 2, 87–107 (1997)MathSciNetCrossRefGoogle Scholar
  32. 72.
    Fernandez-Cara, E., Zuazua, E.: Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. IHP. Anal. Nonlinéaire 17(5), 583–616 (2000)MathSciNetCrossRefGoogle Scholar
  33. 73.
    Fernandez-Cara, E., Guerrero, S.: Global Carleman inequalities for parabolic systems and applications to controllability. SIAM J. Control Optim. 45, 1305–1446 (2006)MathSciNetCrossRefGoogle Scholar
  34. 76.
    Fursikov, A., Imanuvilov, O.Yu.: Controllability of Evolution Equations, Lecture Notes, vol. 34. Seoul National University, Seoul (1996)Google Scholar
  35. 77.
    Gonzales-Burgos, M., Perez-Garcia, R.: Controllability to the trajectories of phase-field models by one control force. Asymptot. Anal. 46, 123–162 (2006)Google Scholar
  36. 78.
    Guerrero, S., Imanuvilov, O.Yu.: Remarks on noncontrollability of the heat equation with memory. ESAIM: Control Optim. Calc. Var. 19, 288–300 (2013)Google Scholar
  37. 81.
    Imanuvilov, O.Yu., Yamamoto, M.: Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semiliner parabolic equations. Publ. Res. Inst. Math. Sci. 39, 227–274 (2003)MathSciNetCrossRefGoogle Scholar
  38. 83.
    Ladyzenskaia, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Translations Mathematical Monographs, vol. 23. American Mathematical Society, Providence (1968)Google Scholar
  39. 85.
    Lebeau, G., Robbiano, L.: Contrôle exact de l’équation de la chaleur. Commun. Part. Differ. Equ. 30, 335–357 (1995)MathSciNetCrossRefGoogle Scholar
  40. 88.
    Le Rousseau, J., Robbiano, L.: Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces. Invent. Math. 183, 245–336 (2011)Google Scholar
  41. 89.
    Lions, J.L.: Quelques Méthodes de resolution des problèmes aux limites Non Linéaires. Dunod, Gauthier-Villars (1969)Google Scholar
  42. 90.
    Liu, Q.: Some results on the controllability of forward stochastic heat equations with control on drift. J. Funct. Anal. 260, 832–851 (2011)Google Scholar
  43. 91.
    Korevaar, N.J., Lewis, J.L.: Convex solutions of certain elliptic equations have constant rank Hessians. Arch. Ration. Mech. Anal. 97(1), 19–32 (1987)MathSciNetCrossRefGoogle Scholar
  44. 93.
    Micu, S., Zuazua, E.: On the lack of null controllability of the heat equation on the half-line. Trans. Am. Math. Soc. 353, 1635–1659 (2000)Google Scholar
  45. 94.
    Micu, S., Zuazua, E.: On the lack of null controllability of the heat equation on the half-space. Part. Math. 58, 1–24 (2001)Google Scholar
  46. 95.
    Miller, L.: Unique continuation estimates for the Laplacian and the heat equation on non-compact manifolds. Math. Res. Lett. 12, 37–47 (2005)MathSciNetCrossRefGoogle Scholar
  47. 96.
    Mizel, V.J., Seidman, T.: Observation and prediction for the heat equation. J. Math. Anal. Appl. 28, 303–312 (1969)MathSciNetCrossRefGoogle Scholar
  48. 102.
    Pandolfi, L.: The controllability of the Gurtin–Pipkin equation: a cosine operator approach. Appl. Math. Optim. 52, 143–165 (2005)MathSciNetCrossRefGoogle Scholar
  49. 103.
    Pandolfi, L.: Riesz systems and controllability of heat equation with memory. Integr. Equ. Oper. Theory 64, 429–453 (2009)MathSciNetCrossRefGoogle Scholar
  50. 105.
    Qin, S., Wang, G.: Controllability of impulse controlled systems of heat equations coupled by constant matrices. J. Differ. Equ. 263(10), 6456–6493 (2017)MathSciNetCrossRefGoogle Scholar
  51. 108.
    Russell, D.L.: A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Stud. Appl. Math. 52, 189–212 (1973)MathSciNetCrossRefGoogle Scholar
  52. 109.
    Russell, D.: Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev. 20(4), 639–739 (1978)MathSciNetCrossRefGoogle Scholar
  53. 110.
    Sirbu, M.: Feedback null controllability of the semilinear heat equation. Differ. Integr. Equ. 15, 115–128 (2002)Google Scholar
  54. 112.
    Tang, S., Zhang, X.: Null controllability for forward and backward stochastic parabolic equations. SIAM J. Control Optim. 48, 2191–2216 (2009)MathSciNetCrossRefGoogle Scholar
  55. 116.
    Tucsnak, M., Weiss, G.: Observations and Control for Operator Semigroups. Birkhäuser, Basel (2009)CrossRefGoogle Scholar
  56. 119.
    Wang, G.: L -controllability for the heat equation and its consequence for the time optimal control problems. SIAM J. Control Optim. 47, 1701–1720 (2008)MathSciNetCrossRefGoogle Scholar
  57. 120.
    Zhang, X.: A unified controllability and observability theory for some stochastic and deterministic partial differential equations. In: Proceedings of the International Congress of Mathematicians, Hyderabad, pp. 3008–3033 (2010)Google Scholar
  58. 121.
    Zuazua, E.: Finite dimensional null controllability of the semilinear heat equations. J. Math. Pures Appl. 76, 237–264 (1997)MathSciNetCrossRefGoogle Scholar
  59. 122.
    Zuazua, E.: Approximate controllability for semilinear heat equation with globally Lipschitz nonlinearities. Control. Cybern. 28, 665–683 (1999)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Viorel Barbu
    • 1

Personalised recommendations