Construction of a Control for the Cubic Semilinear Heat Equation
- 60 Downloads
In this article, we consider the null controllability problem for the cubic semilinear heat equation in bounded domains Ω of ℝ n , 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.
KeywordsNull controllability Cubic semilinear heat equation Linear heat equation
Mathematics Subject Classification (2010)Primary 35K58 Secondary 93B05
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.
- 2.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)Google Scholar
- 4.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)Google Scholar
- 12.Wang, G., Zhang, C.: Observability inequalities from measurable sets for some evolution equations. arXiv:1406.3422 (2014)