Abstract
In this paper we address the problem of null-controllability of heat equations in two different cases: (a) The semilinear heat equation in bounded domains and (b) The linear heat equation in the half line. Concerning the first problem (a) we show that a number of systems in which blow-up arises may be controlled by means of external forces which are localized in an arbitrarily small open set. In the frame of problem (b) we prove that compactly supported initial data may not be driven to zero if the control is supported in a bounded set. This shows that although the velocity of propagation in the heat equation is infinite, this is not sufficient to guarantee null-controllability properties.
We also include a list of open problems.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
S. Anita and V. Barbu, Internal null controllability of nonlinear heat equations, ESA IM: COCV, 5 (2000), 157–173.
S.A. Avdonin and S.A. Ivanov, Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge Univ. Press, 1995.
V. Barbu, Exact controllability of the superlinear heat equation, Appl. Math. Optim., 42(1) (2000), 73–89.
S. Bezerra, Controlabilidade aproximada para a equaçao de calor em dominios nao limitados. Controlabilidade nula, Ph.D. dissertation, Universidade Federal do Rio de Janeiro, July 1999.
T. Cazenave and A. Haraux, Equations d’évolution avec nonlinéarité logarithmique, Ann. Fac. Sci. Toulouse, 2 (1980), 21–51.
T. Cazenave and A. Haraux, Introduction aux problèmes d’évolution semilinéaires, Mathématiques & Applications, Ellipses, Paris 1989.
M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of the heat equation, Nonlinear Analysis TMA,11(10) (1987), 1103–1133.
C. Fabre, J.P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation, Proc. Royal Soc. Edinburgh, 125A (1995), 31–61.
E. Fernández-Cara, Null controllability of the semilinear heat equation, ESA IM: COCV,2 (1997), 87–107.
E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Advances Duff. Eqs., 5 (4–6) (2000), 51–85.
E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing-up semilinear heat equations, Annales de l’IHP,Analyse non-linéaire, to appear.
E. Fernández-Cara and E. Zuazua, Control of weakly blowing up semi-linear heat equations. In Nonlinear PDE’s and Physical Modelling (H. Berestycki and Y. Pomeau, eds.), Kliiwer, NATO-ASI series, to appear.
A.V. 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.
V. Galaktionov, On blow-up and degeneracy for the semilinear heat equation with source, Proc. Royal Soc. Edinburgh,115A, (1990), 19 - 24.
V. Galaktionov and J.L. Vázquez, Regional blow-up in a semilinear heat equation with convergence to a Hamilton-Jacobi equation, SIAM J. Math. Anal., 24(5) (1993), 1254–1276.
J. Henry, Contrôle d’un réacteur enzymatique à l’aide de modèles à paramètres distribués: Quelques problèmes de contrôlabilité de systèmes paraboliques, Ph.D. Thesis, Université Paris VI, 1978.
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 #98–46, University of Tokyo, Graduate School of Mathematics, Komobo, Tokyo, Japan, November 1998.
B.F. Jones., A fundamental solution of the heat equation which is supported in a strip, J. Math. Anal. Appl., 60 (1977), 314–324.
A. Khapalov, Some aspects of the asymptotic behavior of the solutions of the semilinear heat equation and approximate controllability, J. Math. Anal. Appl., 194 (1995), 858–882.
G. Lebeau and L. Robbiano, Contrôle exact de l’équation de la chaleur, Comm. P.D.E., 20 (1995), 335–356.
J.L. Lions, Contrôlabilité exacte, stabilisation et perturbations de systèmes distribués. Tomes 1 ey 2. Masson, RMA 8 & 9, Paris, 1988.
S. Micu and E. Zuazua, On the lack of null-controllability of the heat equation in the half-line, Trans. Amer. Math. Soc., 353(4)(2000), 1635–1659.
S. Micu and E. Zuazua, On the lack of null-controllability of the heat equation in the half-space, Portugaliae Mathematica, 58(1) (2001), 1 - 24.
L. de Teresa and E. Zuazua, Approximate controllability of the heat equation in unbounded domains, Nonlinear Anal., T. M. A., 37(8) (1999), 1059–1090.
K. Yosida, Functional Analysis, Springer Verlag, 1980.
E. Zuazua, Some problems and results on the controllability of Partial Differential Equations, Proceedings of the Second European Conference of Mathematics, Budapest, July 1996, Progress in Mathematics, 169, 1998, Birkhäuser Verlag Basel/Switzerland, pp. 276–311.
E. Zuazua, Controllability of Partial Differential Equations and its Semi-Discrete Approximations. In EDP- Chile (C. Conca et al. eds.), to appear.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Science+Business Media New York
About this chapter
Cite this chapter
Zuazua, E. (2001). Some Results and Open Problems on the Controllability of Linear and Semilinear Heat Equations. In: Colombini, F., Zuily, C. (eds) Carleman Estimates and Applications to Uniqueness and Control Theory. Progress in Nonlinear Differential Equations and Their Applications, vol 46. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0203-5_14
Download citation
DOI: https://doi.org/10.1007/978-1-4612-0203-5_14
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6660-0
Online ISBN: 978-1-4612-0203-5
eBook Packages: Springer Book Archive