We study the global approximate controllability of the one dimensional semilinear convection-diffusion-reaction equation governed in a bounded domain via the coefficient (bilinear control) in the additive reaction term. Clearly, even in the linear case, due to the maximum principle, such system is not globally or locally controllable in any reasonable linear space. It is also well known that for the superlinear terms admitting a power growth at infinity the global approximate controllability by traditional additive controls of localized support is out of question. However, we will show that a system like that can be steered in from any non-negative nonzero initial state into any neighborhood of any desirable non-negative target state by at most three static (-dependent only) above-mentioned bilinear controls, applied subsequently in time, while only one such control is needed in the linear case.
Mots clés : semilinear parabolic equation, global approximate controllability, bilinear control
@article{COCV_2002__7__269_0,
author = {Khapalov, Alexander Y.},
title = {Global non-negative controllability of the semilinear parabolic equation governed by bilinear control},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {269--283},
publisher = {EDP-Sciences},
volume = {7},
year = {2002},
doi = {10.1051/cocv:2002011},
mrnumber = {1925029},
zbl = {1024.93026},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2002011/}
}
TY - JOUR AU - Khapalov, Alexander Y. TI - Global non-negative controllability of the semilinear parabolic equation governed by bilinear control JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2002 SP - 269 EP - 283 VL - 7 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2002011/ DO - 10.1051/cocv:2002011 LA - en ID - COCV_2002__7__269_0 ER -
%0 Journal Article %A Khapalov, Alexander Y. %T Global non-negative controllability of the semilinear parabolic equation governed by bilinear control %J ESAIM: Control, Optimisation and Calculus of Variations %D 2002 %P 269-283 %V 7 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2002011/ %R 10.1051/cocv:2002011 %G en %F COCV_2002__7__269_0
Khapalov, Alexander Y. Global non-negative controllability of the semilinear parabolic equation governed by bilinear control. ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 269-283. doi : 10.1051/cocv:2002011. http://www.numdam.org/articles/10.1051/cocv:2002011/
[1] and, Null controllability of nonlinear convective heat equations. ESAIM: COCV 5 (2000) 157-173. | Numdam | MR | Zbl
[2] , Local Stabilizability of Nonlinear Control Systems. World Scientific, Singapore, Series on Advances in Mathematics and Applied Sciences 8 (1992). | MR | Zbl
[3] and, Feedback stabilization of semilinear control systems. Appl. Math. Opt. 5 (1979) 169-179. | MR | Zbl
[4] and, Nonharmonic Fourier series and the stabilization of distributed semi-linear control systems. Comm. Pure. Appl. Math. 32 (1979) 555-587. | MR | Zbl
[5] , and, Controllability for distributed bilinear systems. SIAM J. Control Optim. (1982) 575-597. | MR | Zbl
[6] , Exact controllability of the superlinear heat equation. Appl. Math. Opt. 42 (2000) 73-89. | MR | Zbl
[7] , and, Bilinear optimal control of the velocity term in a Kirchhoff plate equation. J. Math. Anal. Appl. 238 (1999) 451-467. | MR | Zbl
[8] , Null controllability of the semilinear heat equation. ESAIM: COCV 2 (1997) 87-103. | Numdam | Zbl
[9] and, Controllability for blowing up semilinear parabolic equations. C. R. Acad. Sci. Paris Sér. I Math. 330 (2000) 199-204. | Zbl
[10] , Controllability of some semilnear parabolic problems with multiplicativee control, a talk presented at the Fifth SIAM Conference on Control and its applications, held in San Diego, July 11-14, 2001 (in preparation).
[11] and, Controllability of evolution equations. Res. Inst. Math., GARC, Seoul National University, Lecture Note Ser. 34 (1996). | MR | Zbl
[12] , Étude de la contrôlabilité de certaines équations paraboliques non linéaires, Thèse d'état. Université Paris VI (1978).
[13] , Approximate controllability and its well-posedness for the semilinear reaction-diffusion equation with internal lumped controls. ESAIM: COCV 4 (1999) 83-98. | EuDML | Numdam | MR | Zbl
[14] , Global approximate controllability properties for the semilinear heat equation with superlinear term. Rev. Mat. Complut. 12 (1999) 511-535. | EuDML | MR | Zbl
[15] , A class of globally controllable semilinear heat equations with superlinear terms. J. Math. Anal. Appl. 242 (2000) 271-283. | MR | Zbl
[16] , Bilinear control for global controllability of the semilinear parabolic equations with superlinear terms, in the Special volume “Control of Nonlinear Distributed Parameter Systems”, dedicated to David Russell, Marcel Dekker, Vol. 218 (2001) 139-155. | Zbl
[17] , On bilinear controllability of the parabolic equation with the reaction-diffusion term satisfying Newton's Law, in the special issue of the J. Comput. Appl. Math. dedicated to the memory of J.-L. Lions (to appear). | Zbl
[18] , Controllability of the semilinear parabolic equation governed by a multiplicative control in the reaction term: A qualitative approach, Available as Tech. Rep. 01-7, Washington State University, Department of Mathematics (submitted). | Zbl
[19] , Simultaneous control of a rod equation and a simple Schrödinger equation. Systems Control Lett. 24 (1995) 301-306. | MR | Zbl
[20] , and, Linear and Quasi-linear Equations of Parabolic Type. AMS, Providence, Rhode Island (1968).
[21] , Optimal control of convective-diffusive fluid problem. Math. Models Methods Appl. Sci. 5 (1995) 225-237. | MR | Zbl
[22] , Strong convergence and arbitrarily slow decay of energy for a class of bilinear control problems. J. Differential Equations 81 (1989) 50-67. | MR | Zbl
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