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Approximate controllability and its well-posedness for the semilinear reaction-diffusion equation with internal lumped controls

Published online by Cambridge University Press:  15 August 2002

Alexander Khapalov
Alexander Khapalov*
Affiliation:
Department of Pure and Applied Mathematics, Washington State University, Pullman, WA 99164-3113, USA; [email protected].
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Abstract

We consider the one dimensional semilinear reaction-diffusion equation, governed in Ω = (0,1) by controls, supported on any subinterval of (0, 1), which are the functions of time only. Using an asymptotic approach that we have previously introduced in [9], we show that such a system is approximately controllable at any time in both L2(0,1)( and C0[0,1], provided the nonlinear term f = f(x,t, u) grows at infinity no faster than certain power of log |u|. The latter depends on the regularity and structure of f (x, t, u) in x and t and the choice of the space for controllability. We also show that our results are well-posed in terms of the “actual steering” of the system at hand, even in the case when it admits non-unique solutions.

Keywords

The semilinear reaction-diffusion equationapproximate controllabilityinternal lumped control multiple solutions.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 4 , 1999 , pp. 83 - 98
Copyright
© EDP Sciences, SMAI, 1999

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References

Dolecki, Sz., Observation for the one-dimensional heat equation. Stadia Math. 48 (1973) 291-305. CrossRef
Fabre, C., Uniqueness result for Stokes equations and their consequences in linear and nonlinear problems. ESAIM: Control Optimization and Calculus of Variations 1 (1996) 267-302. CrossRef
Fabre, C., Puel, J.-P. and Zuazua, E., Contrôlabilité approchée de l'équation de la chaleur semi-linéaire. C.R. Acad. Sci. Paris 315 (1992) 807-812.
Fabre, C., Puel, J.-P. and Zuazua, E., Approximate controllability for the semilinear heat equation. Proc. Royal Soc. Edinburg 125A (1995) 31-61. CrossRef
H.O. Fattorini and D.L. Russell, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations. Quarterly of Appl. Mathematics (1974) 45-69.
L.A. Fernández and E. Zuazua, Approximate controllability of the semilinear heat equation via optimal control. JOTA (to appear).
A. Fursikov and O. Imanuvilov, Controllability and of evolution equations. Lect. Note Series 34, Res. Inst. Math., GARC, Seoul National University (1996).
A.Y. Khapalov, On unique continuation of the solutions of the parabolic equation from a curve. Control and Cybernetics, Quarterly 25 (1996) 451-463.
Khapalov, A.Y., 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. CrossRef
O.H. Ladyzhenskaya, V.A. Solonikov and N.N. Ural'ceva, Linear and quasi-linear equations of parabolic type. AMS, Providence, Rhode Island (1968).
J.-L. Lions, Remarques sur la contrôlabilité approchée, in Proc. of ``Jornadas Hispano-Francesas sobre Control de Sistemas Distribuidos'', University of Málaga, Spain (October 1990).
Luxemburg, W.A.J. and Korevaar, J., Entire functions and Müntz-Szász type approximation. Trans. AMS 157 (1971) 23-37.
Mizel, V.J. and Seidman, T.I., Observation and prediction for the heat equation. J. Math. Anal. Appl. 28 (1969) 303-312. CrossRef
F. Rothe, Global Solutions of Reaction-Diffusion Systems. Lecture Notes in Mathematics No. 1072 (Springer-Verlag, Berlin, 1984).
Sakawa, Y., Controllability for partial differential equations of parabolic type. SIAM J. Cont. 12 (1974) 389-400. CrossRef
Seidman, T.I., The coefficient map for certain exponential sums. Neder. Akad. Wetemsch. Indag. Math. 48 (1986) 463-478. CrossRef
Saut, J.-C. and Scheurer, B., Unique continuation for some evolution equations. J. Diff. Equat. 66 (1987) 118-139. CrossRef
L. Schwartz, Étude des sommes d'exponentielles réelles. Actualités Sci. Indust. No. 959 (Hermann, Paris, 1943).
Zhou, H.X., A note on approximate controllability for semilinear one-dimensional heat equation. Appl. Math. Optim. 8 (1982) 275-285.
Zuazua, E., Finite dimensional null controllability for the semilinear heat equation. J. Math. Pures Appl. 76 (1997) 237-264. CrossRef
E. Zuazua, Approximate controllability for semilinear heat equations with globally Lipschitz nonlinearities. Prep. del Depart. de Matematica Applicada, MA-UCM 1998-035, Universidad Complutense de Madrid (1998).