Abstract
The authors prove the global null controllability for the 1-dimensional nonlinear slow diffusion equation by using both a boundary and an internal control. They assume that the internal control is only time dependent. The proof relies on the return method in combination with some local controllability results for nondegenerate equations and rescaling techniques.
Project supported by the ITN FIRST of the Seventh Framework Programme of the European Community (No. 238702), the ERC advanced grant 266907 (CPDENL) of the 7th Research Framework Programme (FP7), DGISPI of Spain (ProjectMTM2011-26119) and the Research Group MOMAT (No. 910480) supported by UCM.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Alt, H.W., Luckhaus, S.: Quasilinear elliptic-parabolic differential equations. Math. Z. 183(3), 311–341 (1983)
Antontsev, S.N., Díaz, J.I., Shmarev, S.: Energy methods for free boundary problems. In: Applications to Nonlinear PDEs and Fluid Mechanics. Progress in Nonlinear Differential Equations and Their Applications, vol. 48. Birkhäuser Boston, Cambridge (2002)
Barenblatt, G.I.: On some unsteady motions of a liquid and gas in a porous medium. Prikl. Mat. Meh. 16, 67–68 (1952)
Beceanu, M.: Local exact controllability of the diffusion equation in one dimension. Abstr. Appl. Anal. 14, 711–793 (2003)
Brezis, H.: Propriétés régularisantes de certains semi-groupes non linéaires. Isr. J. Math. 9, 513–534 (1971)
Brézis, H.: Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations. In: Contributions to Non-linear Functional Analysis. Proc. Sympos., Math. Res., pp. 101–156. Center, Univ. Wisconsin, Madison, Wis (1971). Academic Press, New York, 1971
Brézis, H.: Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert. North Holland, Amsterdam (1973)
Chapouly, M.: Global controllability of nonviscous and viscous Burgers-type equations. SIAM J. Control Optim. 48(3), 1567–1599 (2009)
Coron, J.M.: Global asymptotic stabilization for controllable systems without drift. Math. Control Signals Syst. 5(3), 295–312 (1992)
Coron, J.M.: On the controllability of 2-D incompressible perfect fluids. J. Math. Pures Appl. 75(2), 155–188 (1996)
Coron, J.M.: Control and Nonlinearity. Mathematical Surveys and Monographs, vol. 136. AMS, Providence (2007)
Díaz, G., Díaz, J.I.: Finite extinction time for a class of nonlinear parabolic equations. Commun. Partial Differ. Equ. 4(11), 1213–1231 (1979)
Díaz, J.I., Ramos, Á.M.: Positive and negative approximate controllability results for semilinear parabolic equations. Rev. R. Acad. Cienc. Exactas Fís. Nat. Madr. 89(1–2), 11–30 (1995)
Díaz, J.I., Ramos, Á.M.: Approximate controllability and obstruction phenomena for quasilinear diffusion equations. In: Bristeau, M.-O., Etgen, G., Fitzgibbon, W., et al. (eds.) Computational Science for the 21st Century, pp. 698–707. Wiley, Chichester (1997)
Díaz, J.I., Ramos, Á.M.: Un método de viscosidad para la controlabilidad aproximada de ciertas ecuaciones parabólicas cuasilineales. In: Caraballo, T., et al. (eds.) Actas de Jornada Científica en Homenaje al Prof. A. Valle Sánchez, pp. 133–151. Universidad de Sevilla (1997)
Evans, L.C.: Partial Differential Equations, 2nd edn. Graduate Studies in Mathematics, vol. 19. AMS, Providence (2010)
Fabre, C., Puel, J.P., Zuazua, E.: Approximate controllability of the semilinear heat equation. Proc. R. Soc. Edinb. A 125(1), 31–61 (1995)
Filo, J.: A nonlinear diffusion equation with nonlinear boundary conditions: method of lines. Math. Slovaca 38(3), 273–296 (1988)
Fursikov, A.V., Imanuvilov, O.Y.: Controllability of evolution equations. Lecture Notes Series, vol. 34. Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul (1996)
Henry, J.: Etude de la contrôlabilité de certains équations paraboliques, Thèse d’ État, Université de Paris VI, Paris (1978)
Lions, J.L.: Quelques Méthodes de Résolution des Probl‘emes aux Limites non Linéaires. Dunod, Paris (1969)
Marbach, F.: Fast global null controllability for a viscous Burgers equation despite the presence of a boundary layer (2013). Preprint. arXiv:1301.2987v1
Vázquez, J.L.: The Porous Medium Equation. Oxford Mathematical Monographs, Mathematical Theory. Clarendon/Oxford University Press, Oxford (2007)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Coron, JM., Díaz, J.I., Drici, A., Mingazzini, T. (2014). Global Null Controllability of the 1-Dimensional Nonlinear Slow Diffusion Equation. In: Ciarlet, P., Li, T., Maday, Y. (eds) Partial Differential Equations: Theory, Control and Approximation. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41401-5_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-41401-5_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-41400-8
Online ISBN: 978-3-642-41401-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)