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Null controllability of some reaction-diffusion systems with only one control force in moving domains. (English) Zbl 1333.93049
Summary: In this article, the authors establish the local null controllability property for semilinear parabolic systems in a domain whose boundary moves in time by a single control force acting on a prescribed subdomain. The proof is based on Kakutani’s fixed point theorem combined with observability estimates for the associated linearized system.

93B05 Controllability
35K57 Reaction-diffusion equations
Full Text: DOI
[1] Benabdallah, A.; Naso, M. G., Null controllability of a thermoelastic plate, Abstr. Appl. Anal, 7, 585-599, (2002) · Zbl 1013.35008
[2] Bernardi, M. L.; Bonfanti, G.; Lutteroti, F., Abstract Schrödinger type differential equations with variable domain, J. Math. Ann. and Appl, 211, 84-105, (1997) · Zbl 0885.35024
[3] Cabanillas, V. R.; Menezes, S. B.; Zuazua, E., Null controllability in unbounded domains for the semilinear heat equation with nonlinearities involving gradient terms, J. Optm. Theory Appl, 110, 245-264, (2001) · Zbl 0997.93048
[4] Chen, G. Q.; Frid, H., Divergence-measure fields and hyperbolic conservation laws, Archive Rat. Mech. Anal, 147, 89-118, (1999) · Zbl 0942.35111
[5] Cooper, J.; Bardos, C., A nonlinear wave equation in a time dependent domain, J. Math. Anal. Appl, 42, 29-60, (1973) · Zbl 0265.35050
[6] Menezes, S. B.; Limaco, J.; Medeiros, L. A., Remarks on null controllability for semilinear heat equation in moving domains, Eletronic J. of Qualitative Theory of Differential Equations, 16, 1-32, (2003) · Zbl 1032.35096
[7] Doubova, A.; Fernández-Cara, E.; González-Burgos, M.; Zuazua, E., On the controllability of parabolic systems with a nonlinear term involving the state and the gradient, SIAM J. Control Optim, 41, 798-819, (2002) · Zbl 1038.93041
[8] Fabre, C.; Puel, J. P.; Zuazua, E., Approximate controllability of the semilinear heat equation, Proc. Royal Soc. Edinburgh, Ser. A, 125, 31-61, (1995) · Zbl 0818.93032
[9] Fernández-Cara, E.; Guerrero, S., Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim, 45, 1395-1446, (2006) · Zbl 1121.35017
[10] Fernández-Cara, E.; Zuazua, E., Null and approximate controllability of weakly blowing-up semilinear heat equations, ann. inst. Henri Poincaré, Analyse non Linéaire, 17, 583-616, (2000) · Zbl 0970.93023
[11] Fursikov, A. and Imanuvilov, O., Controllability of evolution equations, Lecture Notes, Vol. 34, Seoul National University, Korea, 1996. · Zbl 0862.49004
[12] González-Burgos, M.; Pérez-García, R., Controllability results for some nonlinear coupled parabolic systems by one control force, Asymptot. Anal, 46, 123-162, (2006) · Zbl 1124.35026
[13] He, C.; Hsiano, L., Two dimensional Euler equations in a time dependent domain, J. Diff. Equations, 163, 265-291, (2000) · Zbl 0958.35096
[14] Imanuvilov, O.Y.u., Controllability of parabolic equations (in Russian), Mat. Sbornik. Novaya Seriya, 186, 109-132, (1995)
[15] Inoue, A., Sur □ut + u3 = f dans un domaine non-cylindrique, J. Math. Anal. Appl, 46, 777-819, (1970) · Zbl 0282.35060
[16] Khodja, F.; Benabdallah, A.; Dupaix, C., Null controllability of some reaction-diffusion system with one control force, J. Math. Anal. and Appl, 320, 928-943, (2006) · Zbl 1157.93004
[17] Khodja, F.; Benabdallah, A.; Dupaix, C.; Kostin, I., Controllability to the trajectories of phase-field models by one control force, SIAM J. Control Optim, 42, 1661-1680, (2003) · Zbl 1052.35080
[18] Lions, J. L., Quelques méthodes de résolution des probl‘emes aux limites non-linéaires, Dunod, Paris, 1960.
[19] Lions, J. L., Une remarque sur LES problèmes d’evolution nonlineaires dans LES domaines non cylindriques (in French), Rev. Roumaine Math. Pures Appl, 9, 11-18, (1964) · Zbl 0178.12302
[20] Medeiros, L. A., Non-linear wave equations in domains with variable boundary, Arch. Rational Mech. Anal, 47, 47-58, (1972) · Zbl 0239.35066
[21] Miranda, M. M.; Limaco, J., The Navier-Stokes equation in noncylindrical domain, Comput. Appl. Math, 16, 247-265, (1997) · Zbl 0895.35076
[22] Miranda, M. M.; Medeiros, L. A., Contrôlabilité exacte de l’équation de Schrödinger dans des domaines non cylindriques, C. R. Acad. Sci. Paris, 319, 685-689, (1994) · Zbl 0809.93026
[23] Nakao, M.; Narazaki, T., Existence and decay of some nonlinear wave equations in noncylindrical domains, Math. Rep, XI-2, 117-125, (1978)
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