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Exact controllability of nonlinear diffusion equations arising in reactor dynamics. (English) Zbl 1156.93321
Summary: This paper studies the problems of local exact controllability of nonlinear and global exact null controllability of linear parabolic integro-differential equations, respectively, with mixed and Neumann boundary data with distributed controls acting on a subdomain $\omega $ of $\Omega \subset \bbfR^n$. The proof of the linear problem relies on a Carleman-type estimate and observability inequality for the adjoint equations and that the nonlinear one, on the fixed point technique.

MSC:
93B05Controllability
93C20Control systems governed by PDE
45K05Integro-partial differential equations
35K50Systems of parabolic equations, boundary value problems (MSC2000)
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References:
[1] Adams, R. A.; Fournier, J. F.: Sobolev spaces, (2003) · Zbl 1098.46001
[2] Akcasu, Z.; Lellouche, G. S.; Shotkin, L. M.: Mathematical methods in nuclear reactor dynamics, (1971)
[3] Balachandran, K.; Dauer, J. P.: Controllability of nonlinear systems in Banach spaces; a survey, J. optim. Theory appl. 115, 7-28 (2002) · Zbl 1023.93010 · doi:10.1023/A:1019668728098
[4] Barbu, V.: Controllability of parabolic and Navier -- Stokes equations, Sci. math. Japon. 56, 143-211 (2002) · Zbl 1010.93054
[5] Barbu, V.; Havarneanu, T.; Popa, C.; Sritharan, S. S.: Exact controllability for the magnetohydrodynamic equations, Commun. pure appl. Math. 54, 732-783 (2003) · Zbl 1121.93306 · doi:10.1002/cpa.10072
[6] Barbu, V.; Sritharan, S. S.: Flow invariance preserving feedback controllers for the Navier -- Stokes equations, J. math. Anal. appl. 255, 281-307 (2001) · Zbl 1073.93030 · doi:10.1006/jmaa.2000.7256
[7] Bardos, C.; Lebeau, G.; Rauch, J.: Controle et stabilisation de l’equation des ondes, SIAM J. Control optim. 30, 1024-1065 (1992)
[8] Chae, D.; Imanivilov, O. Yu.; Kim, M. S.: Exact controllability for semilinear parabolic equations with Neumann boundary conditions, J. dyn. Control syst. 2, 449-483 (1996) · Zbl 0946.93007 · doi:10.1007/BF02254698
[9] Fursikov, A. V.; Imanuvilov, O. Yu.: Controllability of evolution equations, lecture notes series, Controllability of evolution equations, lecture notes series 34 (1996) · Zbl 0862.49004
[10] Fursikov, A. V.; Imanuvilov, O. Yu.: Exact controllability of the Navier -- Stokes and Boussinesq equations, Russian math. Surv. 54, 565-618 (1999) · Zbl 0970.35116 · doi:10.1070/rm1999v054n03ABEH000153
[11] Hörmander, L.: Linear partial differential operators, Linear partial differential operators (1985) · Zbl 0601.35001
[12] Imanuvilov, O. Yu.: Boundary controllability of parabolic equations, Sb. math. 186, 879-900 (1995)
[13] Imanuvilov, O. Yu.: Remarks on controllability of Navier -- Stokes equations, ESAIM control optim. Calculus variations 6, 49-97 (2001) · Zbl 0961.35104
[14] Imanuvilov, O. Yu.; Yamamoto, M.: Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Publ. RIMS Kyoto univ. 39, 227-274 (2003) · Zbl 1065.35079 · doi:10.2977/prims/1145476103
[15] Kastenberg, W. E.; Chambre, P. L.: On the stability of nonlinear space-dependent reactor kinetics, Nucl. sci. Eng. 31, 67-79 (1968)
[16] Komornik, V.: Exact controllability and stabilization, the multiplier method, (1995) · Zbl 0937.93003
[17] Ladyzenskaya, O. A.: The mathematical theory of viscous incompressible flow, (1969) · Zbl 0184.52603
[18] Levin, J. J.; Nohel, J. A.: The integro-differential equations of a class of nuclear reactors with delayed neutrons, Arch. rational mech. Anal. 31, 151-171 (1968) · Zbl 0167.41303 · doi:10.1007/BF00281375
[19] Exacte, J. L. Lions Controlabilité: Stabilisation et perturbations the systémes distribués, tome 1 and tome 2, (1988)
[20] Pachpatte, B. G.: On a nonlinear diffusion system arising in reactor dynamics, J. math. Anal. appl. 94, 501-508 (1983) · Zbl 0524.35055 · doi:10.1016/0022-247X(83)90078-1
[21] Pao, C. V.: Solution of a nonlinear integro-differential system arising in nuclear reactor dynamics, J. math. Anal. appl. 48, 470-492 (1974) · Zbl 0293.45016 · doi:10.1016/0022-247X(74)90171-1
[22] Pao, C. V.: Bifurcation analysis of a nonlinear diffusion system in reactor dynamics, Appl. anal. 9, 107-119 (1979) · Zbl 0404.45009 · doi:10.1080/00036817908839258
[23] Sakthivel, K.; Balachandran, K.; Lavanya, R.: Exact controllability of partial integro-differential equations with mixed boundary conditions, J. math. Anal. appl. 325, 1257-1279 (2007) · Zbl 1110.93009 · doi:10.1016/j.jmaa.2006.02.034
[24] K. Sakthivel, K. Balachandran, B.R.Nagaraj, On a class of nonlinear parabolic control systems with memory effects, Int. J. Control, in press. · Zbl 1152.93312 · doi:10.1080/00207170701447114
[25] Tataru, D.: Boundary controllability for conservative partial differential equations, Appl. math. Optim. 31, 257-295 (1995) · Zbl 0836.35085 · doi:10.1007/BF01215993
[26] Tataru, D.: Carleman estimates and unique continuation and controllability for anisotropic partial differential equations, Contemp. math. 209, 267-279 (1997) · Zbl 0906.35017
[27] Temam, R.: Navier -- Stokes equations and nonlinear functional analysis, (1983) · Zbl 0522.35002
[28] Vrabie, I. I.: Compactness methods for nonlinear evolutions, (1995) · Zbl 0842.47040
[29] Yanik, E. G.; Fairweather, G.: Finite element methods for parabolic and hyperbolic partial integro-differential equations, Nonlinear anal. 12, 785-809 (1988) · Zbl 0657.65142 · doi:10.1016/0362-546X(88)90039-9