Approximate controllability for the semilinear heat equation involving gradient terms. (English) Zbl 0952.49003

From the summary: “The paper deals with the approximate controllability of the semilinear heat equation, when the nonlinear term depends on both the state \(y\) and its spatial gradient \(\nabla y\) and the control acts on a nonempty open subset of the domain. The proof relies on the fact that the nonlinearity is globally Lipschitz with respect to \((y,\nabla y)\). The approximate controllability is viewed as the limit of a sequence of optimal control problems. It is proved that approximate controllability can be obtained simultaneously with exact controllability over finite-dimensional subspaces”.


49J20 Existence theories for optimal control problems involving partial differential equations
Full Text: DOI


[1] Fabre, C., Uniqueness Results for Stokes Equations and Their Consequences in Linear and Nonlinear Control Problems, European Series in Applied and Industrial Mathematics: Contrôle, Optimisation et Calcul des Variations, Vol. 1, pp. 267–302, 1996. · Zbl 0872.93039
[2] Lions, J. L., Contrôle Optimal de Systèmes Gouvernés par des Equations aux Dérivées Partielles, Dunod, Paris, France, 1968.
[3] Henry, J., Contrôle d’un Réacteur Enzymatique à l’Aide de Modèles à Paramètres Distribués: Quelques Problèmes de Contrôlabilité de Systèmes Paraboliques, PhD Thesis, Université de Paris VI, 1978.
[4] Fabre, C., Puel, J. P., and Zuazua, E., Approximate Controllability of the Semilinear Heat Equation, Proceedings of the Royal Society of Edinburgh, Section A, Vol. 125, pp. 31–61, 1995. · Zbl 0818.93032
[5] Lions, J. L., Remarques sur la Contrôlabilité Approchée, Jornadas Hispano-Francesas sobre Control de Sistemas Distribuídos, University of Málaga, Málaga, Spain, pp. 77–87, 1991.
[6] Fursikov, A. V., and Imanuvilov, O. Y., On Approximate Controllability of the Stokes System, Annales de la Faculté des Sciences de Toulouse, Vol. 2, pp. 205–232, 1993. · Zbl 0925.93416
[7] Zuazua, E., Finite-Dimensional Null Controllability for the Semilinear Heat Equation, Journal de Mathématiques Pures et Appliquées, Vol. 76, pp. 237–264, 1997. · Zbl 0872.93014
[8] Dautray, R., and Lions, J. L., Mathematical Analysis and Numerical Methods for Science and Technology, Springer Verlag, Berlin, Germany, 1992. · Zbl 0755.35001
[9] Ladyzhenskaya, O. A., Solonnikov, V. A., and Ural’tseva, N. N., Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, Rhode Island, 1968.
[10] Ladyzhenskaya, O. A., Boundary-Value Problems of Mathematical Physics, Springer Verlag, New York, New York, 1985. · Zbl 0588.35003
[11] Aubin, J. P., Un Théorème de Compacité, Comptes Rendus de l’Académie des Sciences de Paris, Vol. 256, pp. 5042–5044, 1963. · Zbl 0195.13002
[12] Lions, J. L., and Magenes, E., Problèmes aux Limites non Homogènes, Dunod, Paris, France, 1968. · Zbl 0101.07901
[13] Mizohata, S., Unicité du Prolongement des Solutions pour Quelques Opérateurs Différentiels Paraboliques, Memoirs of the College of Science, University of Kyoto, Series A, Vol. 31, pp. 219–239, 1958.
[14] Saut, J. C., and Scheurer, B., Unique Continuation for Some Evolution Equations, Journal of Differential Equations, Vol. 66, pp. 118–139, 1987. · Zbl 0631.35044
[15] Pourciau, B. H., Modern Multiplier Rules, American Mathematical Monthly, Vol. 87, pp. 433–452, 1980. · Zbl 0454.90067
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.