Global non-negative controllability of the semilinear parabolic equation governed by bilinear control

Alexander Y. Khapalov

ESAIM: Control, Optimisation and Calculus of Variations (2002)

  • Volume: 7, page 269-283
  • ISSN: 1292-8119

Abstract

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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 L 2 ( 0 , 1 ) from any non-negative nonzero initial state into any neighborhood of any desirable non-negative target state by at most three static ( x -dependent only) above-mentioned bilinear controls, applied subsequently in time, while only one such control is needed in the linear case.

How to cite

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Khapalov, Alexander Y.. "Global non-negative controllability of the semilinear parabolic equation governed by bilinear control." ESAIM: Control, Optimisation and Calculus of Variations 7 (2002): 269-283. <http://eudml.org/doc/245797>.

@article{Khapalov2002,
abstract = {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 $ L^2 (0,1)$ from any non-negative nonzero initial state into any neighborhood of any desirable non-negative target state by at most three static ($x$-dependent only) above-mentioned bilinear controls, applied subsequently in time, while only one such control is needed in the linear case.},
author = {Khapalov, Alexander Y.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {semilinear parabolic equation; global approximate controllability; bilinear control; bilinear system; reachability},
language = {eng},
pages = {269-283},
publisher = {EDP-Sciences},
title = {Global non-negative controllability of the semilinear parabolic equation governed by bilinear control},
url = {http://eudml.org/doc/245797},
volume = {7},
year = {2002},
}

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
PB - EDP-Sciences
VL - 7
SP - 269
EP - 283
AB - 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 $ L^2 (0,1)$ from any non-negative nonzero initial state into any neighborhood of any desirable non-negative target state by at most three static ($x$-dependent only) above-mentioned bilinear controls, applied subsequently in time, while only one such control is needed in the linear case.
LA - eng
KW - semilinear parabolic equation; global approximate controllability; bilinear control; bilinear system; reachability
UR - http://eudml.org/doc/245797
ER -

References

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  1. [1] S. Anita and V. Barbu, Null controllability of nonlinear convective heat equations. ESAIM: COCV 5 (2000) 157-173. Zbl0938.93008MR1744610
  2. [2] A. Baciotti, Local Stabilizability of Nonlinear Control Systems. World Scientific, Singapore, Series on Advances in Mathematics and Applied Sciences 8 (1992). Zbl0757.93061MR1148363
  3. [3] J.M. Ball and M. Slemrod, Feedback stabilization of semilinear control systems. Appl. Math. Opt. 5 (1979) 169-179. Zbl0405.93030MR533618
  4. [4] J.M. Ball and M. Slemrod, Nonharmonic Fourier series and the stabilization of distributed semi-linear control systems. Comm. Pure. Appl. Math. 32 (1979) 555-587. Zbl0394.93041MR528632
  5. [5] J.M. Ball, J.E. Mardsen and M. Slemrod, Controllability for distributed bilinear systems. SIAM J. Control Optim. (1982) 575-597. Zbl0485.93015MR661034
  6. [6] V. Barbu, Exact controllability of the superlinear heat equation. Appl. Math. Opt. 42 (2000) 73-89. Zbl0964.93046MR1751309
  7. [7] M.E. Bradley, S. Lenhart and J. Yong, Bilinear optimal control of the velocity term in a Kirchhoff plate equation. J. Math. Anal. Appl. 238 (1999) 451-467. Zbl0936.49003MR1715493
  8. [8] E. Fernández–Cara, Null controllability of the semilinear heat equation. ESAIM: COCV 2 (1997) 87-103. Zbl0897.93011
  9. [9] E. Fernández–Cara and E. Zuazua, Controllability for blowing up semilinear parabolic equations. C. R. Acad. Sci. Paris Sér. I Math. 330 (2000) 199-204. Zbl0952.93061
  10. [10] L.A. Fernández, 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. [11] A. Fursikov and O. Imanuvilov, Controllability of evolution equations. Res. Inst. Math., GARC, Seoul National University, Lecture Note Ser. 34 (1996). Zbl0862.49004MR1406566
  12. [12] J. Henry, Étude de la contrôlabilité de certaines équations paraboliques non linéaires, Thèse d’état. Université Paris VI (1978). 
  13. [13] A.Y. Khapalov, Approximate controllability and its well-posedness for the semilinear reaction-diffusion equation with internal lumped controls. ESAIM: COCV 4 (1999) 83-98. Zbl0926.93007MR1680760
  14. [14] A.Y. Khapalov, Global approximate controllability properties for the semilinear heat equation with superlinear term. Rev. Mat. Complut. 12 (1999) 511-535. Zbl1006.35014MR1740472
  15. [15] A.Y. Khapalov, A class of globally controllable semilinear heat equations with superlinear terms. J. Math. Anal. Appl. 242 (2000) 271-283. Zbl0951.35062MR1737850
  16. [16] A.Y. Khapalov, 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. Zbl0983.93023
  17. [17] A.Y. Khapalov, 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). Zbl1119.93017
  18. [18] A.Y. Khapalov, 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). Zbl1041.93026
  19. [19] K. Kime, Simultaneous control of a rod equation and a simple Schrödinger equation. Systems Control Lett. 24 (1995) 301-306. Zbl0877.93003MR1321139
  20. [20] O.H. Ladyzhenskaya, V.A. Solonikov and N.N. Ural’ceva, Linear and Quasi-linear Equations of Parabolic Type. AMS, Providence, Rhode Island (1968). 
  21. [21] S. Lenhart, Optimal control of convective-diffusive fluid problem. Math. Models Methods Appl. Sci. 5 (1995) 225-237. Zbl0828.76066MR1321328
  22. [22] S. Müller, Strong convergence and arbitrarily slow decay of energy for a class of bilinear control problems. J. Differential Equations 81 (1989) 50-67. Zbl0711.35017MR1012199

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