Abstract
In this article, we consider the null controllability problem for the cubic semilinear heat equation in bounded domains Ω of ℝn, n ≥ 3 with Dirichlet boundary conditions for small initial data. A constructive way to compute a control function acting on any nonempty open subset ω of Ω is given such that the corresponding solution of the cubic semilinear heat equation can be driven to zero at a given final time T. Furthermore, we provide a quantitative estimate for the smallness of the size of the initial data with respect to T that ensures the null controllability property.
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References
Aniţa, S., Tataru, D.: Null controllability for the dissipative semilinear heat equation. Appl. Math. Optim. 46, 97–105 (2002)
Cazenave, T., Haraux, A.: An Introduction to Semilinear Evolution Equations. Oxford Lecture Series in Mathematics and its Applications, vol. 13. The Clarendon Press, Oxford University Press, New York (1998)
Coron, J.-M., Guerrero, S., Rosier, L.: Null controllability of a parabolic system with a cubic coupling term. SIAM J. Control Optim. 48, 5629–5652 (2010)
Díaz, J.I., Lions, J.L.: On the approximate controllability for some explosive parabolic problems. In: Hoffmann, K.-H., et al. (eds.) Optimal Control of Partial Differential Equations (Chemnitz, 1998), Internat. Ser. Numer. Math., vol. 133, pp 115–132. Basel, Birkhäuser (1999)
Fernández-Cara, E., Guerrero, S.: Global Carleman inequalities for parabolic systems and applications to controllability. SIAM J. Control Optim. 45, 1395–1446 (2006)
Fernández-Cara, E., Zuazua, E.: Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. Henri Poincaré Anal. Non Linéaire 17, 583–616 (2000)
Fursikov, A.V., Yu Imanuvilov, O.: Controllability of Evolution Equations. Lecture Notes, vol. 34. Seoul National University, Seoul (1996)
Lebeau, G., Robbiano, R.: Contróle exact de l’équation de la chaleur. Commun. Partial Differ. Equ. 20, 335–356 (1995)
Liu, Y., Takahashi, T., Tucsnak, M.: Single input controllability of a simplified fluid-structure interaction model. ESAIM Control Optim. Calc. Var. 19, 20–42 (2013)
Phung, K.D., Wang, G.: An observability estimate for parabolic equations from a measurable set in time and its applications. J. Eur. Math. Soc. 15, 681–703 (2013)
Phung, K.D., Wang, L., Zhang, C.: Bang-bang property for time optimal control of semilinear heat equation. Ann. Inst. Henri Poincaré Anal. Non Linéaire 31, 477–499 (2014)
Wang, G., Zhang, C.: Observability inequalities from measurable sets for some evolution equations. arXiv:1406.3422 (2014)
Zuazua, E.: Controllability and observability of partial differential equations: some results and open problems. In: Dafermos, C.M., Pokorný, M (eds.) Handbook of Differential Equations: Evolutionary Equations, vol. 3, pp 527–621. Elsevier/North-Holland, Amsterdam (2007)
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The author would like to express her gratitude to both referees of this journal for the valuable comments, important suggestions, and corrections of this work which improved substantially the first version of this article.
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This work was written while the author was visiting the University of Orleans (France). She thanks the MAPMO department of mathematics of the University of Orleans. The author also wishes to acknowledge Region Centre for its financial support.
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Vo, T.M.N. Construction of a Control for the Cubic Semilinear Heat Equation. Vietnam J. Math. 44, 587–601 (2016). https://doi.org/10.1007/s10013-015-0171-x
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DOI: https://doi.org/10.1007/s10013-015-0171-x