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

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

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

## Access Full Article

top## Abstract

top## How to cite

topKhapalov, 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

top- [1] S. Anita and V. Barbu, Null controllability of nonlinear convective heat equations. ESAIM: COCV 5 (2000) 157-173. Zbl0938.93008MR1744610
- [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] J.M. Ball and M. Slemrod, Feedback stabilization of semilinear control systems. Appl. Math. Opt. 5 (1979) 169-179. Zbl0405.93030MR533618
- [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] J.M. Ball, J.E. Mardsen and M. Slemrod, Controllability for distributed bilinear systems. SIAM J. Control Optim. (1982) 575-597. Zbl0485.93015MR661034
- [6] V. Barbu, Exact controllability of the superlinear heat equation. Appl. Math. Opt. 42 (2000) 73-89. Zbl0964.93046MR1751309
- [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] E. Fernández–Cara, Null controllability of the semilinear heat equation. ESAIM: COCV 2 (1997) 87-103. Zbl0897.93011
- [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] 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] A. Fursikov and O. Imanuvilov, Controllability of evolution equations. Res. Inst. Math., GARC, Seoul National University, Lecture Note Ser. 34 (1996). Zbl0862.49004MR1406566
- [12] J. Henry, Étude de la contrôlabilité de certaines équations paraboliques non linéaires, Thèse d’état. Université Paris VI (1978).
- [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] 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] 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] 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] 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] 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] K. Kime, Simultaneous control of a rod equation and a simple Schrödinger equation. Systems Control Lett. 24 (1995) 301-306. Zbl0877.93003MR1321139
- [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] S. Lenhart, Optimal control of convective-diffusive fluid problem. Math. Models Methods Appl. Sci. 5 (1995) 225-237. Zbl0828.76066MR1321328
- [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

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.