Exact controllability of the superlinear heat equation

Xu Youjun(xuyoujun7618yahoo.com.cn)
Liu Zhenhai (zhhliumail.csu.edu.cn)

Abstract: In this paper, we consider the controllability of a semilinear heat equation with a nonlinearity that has a superlinear growth at infinity with Dirichlet boundary conditions in a bounded domain of $ \mathbb{R}^{N}$. The proof of the main result in this paper involve such inequalities and rely on the study of these linear problems and appropriate fixed point arguments.

Keywords: Controllability; parabolic equation; superlinearity.

Category 1: Applications -- Science and Engineering (Control Applications )

Category 2: Infinite Dimensional Optimization (Distributed Control )

Category 3: Applications -- Science and Engineering (Optimization of Systems modeled by PDEs )

Citation: [1] V. Barbu, Exact controllability of the superlinear heat equation. Appl. Math. Opt. 42 (2000) 73-89. [2] Fern¶andez-Cara and E. Zuazua, Controllability for blowing up semilinear parabolic equations. C. R. Acad. Sci. Paris S¶er. I Math. 330 (2000) 199-204. [3] C. Fabre, J.-P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh Sect. A 125 (1995) 31-61. [4] A. Doubova, E. Fern¶andez-Cara, M. Gonza¶aez-Burgos and E. Zuazua, On the control- lability of parabolic system with a nonlinear term involving the state and the gradient. SIAM J. Control Optim, 41 (3) (2003) 798-819. [5] S. Anita and V. Barbu, Null controllability of nonlinear convective heat equation. ESAIM: COCV 5 (2000) 157-173. [6] Bodart, O., Gonza¶aez-Burgos, M., Pe¶eez-Garc a, R. Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient. Nonlinear Anal. 57 (5-6) (2004) 687-711. [7] Bodart, O., Gonza¶aez-Burgos, M., Pe¶eez-Garc a, R. A local result on insensitizing controls for a semilinear heat equation with nonlinear boundary Fourier conditions SIAM J. Control Optim, 43 (3) (2004) 955-969. [8] Bodart, O., Gonza¶aez-Burgos, M., Pe¶eez-Garc a, R. Existence of Insensitizing Controls for a Semilinear Heat Equation with a Superlinear Nonlinearity. Communications in Partial Di®erential Equations. 29 (7-8)(2004) 1017-1050. [9] Fern¶andez-Cara E., Null controllability of the semilinear heat equation. ESAIM: COCV 2 (1997) 87-107. [10] Fern¶andez-Cara E., Zuazua E., Null and approximate controllability for weakly blow- ing up semilinear heat equations. Ann. Inst. Henri Poincar¶e, Analyse non lin¶eaire 17, 5 (2000) 583-616. [11] A. Fursikov, O. Yu. Imanuvilov, Controllability of Evolution equations, in: Lecture Notes Series, Vol. 34, Seoul National University, seoul, 1996. [12] V. Barbu, Controllability of parabolic and Navier-Stokes equations, Scientia Mathe- matica Japonica, 6 (2002) 143-211. [13] A. Doubova, A. Osses, J.-P. Puel, Exact controllability to trajectories for semilinear heat equations with discontinuous coe±cients, ESAIM:COCV, 8 (2002) 621-661. [14] X. Zhang, Exact Controllability of Semi-linear Distributed Parameter Systems, Chi- nese Higher Education Press, Beijing, 2004. [15] F.B. Weissler, Local existence and nonexistence for semilinear parabolic equations in Lp, Indiana Univ. Math. J. 29 (1980), no. 1, 79-102. [16] F.B. Weissler, Semilinear evolution equations in Banach spaces, J. Funct. Anal. 32 (1979), no. 3, 277-296. [17] E. Zuazua, Approximate controllability for semilinear heat equations with globally Lips- chitz nonlinearities, Control and Cybernetics, 28 (1999), no. 3, 665-683.

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Entry Submitted: 05/11/2008
Entry Accepted: 05/11/2008
Entry Last Modified: 05/11/2008

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