# zbMATH — the first resource for mathematics

Global non-negative controllability of the semilinear parabolic equation governed by bilinear control. (English) Zbl 1024.93026
Summary: 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 a 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.

##### MSC:
 93C20 Control/observation systems governed by partial differential equations 93B03 Attainable sets, reachability
Full Text:
##### References:
 [1] S. Anita and V. Barbu , Null controllability of nonlinear convective heat equations . ESAIM: COCV 5 ( 2000 ) 157 - 173 . Numdam | MR 1744610 | Zbl 0938.93008 · Zbl 0938.93008 [2] A. Baciotti , Local Stabilizability of Nonlinear Control Systems . World Scientific, Singapore, Series on Advances in Mathematics and Applied Sciences 8 ( 1992 ). MR 1148363 | Zbl 0757.93061 · Zbl 0757.93061 [3] J.M. Ball and M. Slemrod , Feedback stabilization of semilinear control systems . Appl. Math. Opt. 5 ( 1979 ) 169 - 179 . MR 533618 | Zbl 0405.93030 · Zbl 0405.93030 [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 . MR 528632 | Zbl 0394.93041 · Zbl 0394.93041 [5] J.M. Ball , J.E. Mardsen and M. Slemrod , Controllability for distributed bilinear systems . SIAM J. Control Optim. ( 1982 ) 575 - 597 . MR 661034 | Zbl 0485.93015 · Zbl 0485.93015 [6] V. Barbu , Exact controllability of the superlinear heat equation . Appl. Math. Opt. 42 ( 2000 ) 73 - 89 . MR 1751309 | Zbl 0964.93046 · Zbl 0964.93046 [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 . MR 1715493 | Zbl 0936.49003 · Zbl 0936.49003 [8] E. Fernández-Cara , Null controllability of the semilinear heat equation . ESAIM: COCV 2 ( 1997 ) 87 - 103 . Numdam | Zbl 0897.93011 · Zbl 0897.93011 [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 . Zbl 0952.93061 · Zbl 0952.93061 [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 ). MR 1406566 | Zbl 0862.49004 · Zbl 0862.49004 [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 . Numdam | MR 1680760 | Zbl 0926.93007 · Zbl 0926.93007 [14] A.Y. Khapalov , Global approximate controllability properties for the semilinear heat equation with superlinear term . Rev. Mat. Complut. 12 ( 1999 ) 511 - 535 . MR 1740472 | Zbl 1006.35014 · Zbl 1006.35014 [15] A.Y. Khapalov , A class of globally controllable semilinear heat equations with superlinear terms . J. Math. Anal. Appl. 242 ( 2000 ) 271 - 283 . MR 1737850 | Zbl 0951.35062 · Zbl 0951.35062 [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 . Zbl 0983.93023 · Zbl 0983.93023 [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). Zbl 1119.93017 · Zbl 1119.93017 [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). Zbl 1041.93026 · Zbl 1041.93026 [19] K. Kime , Simultaneous control of a rod equation and a simple Schrödinger equation . Systems Control Lett. 24 ( 1995 ) 301 - 306 . MR 1321139 | Zbl 0877.93003 · Zbl 0877.93003 [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 ). · Zbl 0174.15403 [21] S. Lenhart , Optimal control of convective-diffusive fluid problem . Math. Models Methods Appl. Sci. 5 ( 1995 ) 225 - 237 . MR 1321328 | Zbl 0828.76066 · Zbl 0828.76066 [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 . MR 1012199 | Zbl 0711.35017 · Zbl 0711.35017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.