Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients. (English) Zbl 1092.93006
The authors investigate the exact controllability to trajectories for a transmission problem for semilinear heat equation. Transmission conditions on the interface and Dirichlet boundary conditions at the external part of the boundary are given. The system can be viewed as a semilinear heat equation with discontinuous diffusion coefficients. If the nonlinear term grows (at infinity) slower than \(|r|\log^{3/2}(1=|r|)\), then the exact controllability to trajectories is proved for the case when the control acts in the region with the smaller diffusion coefficient. The proof is based on the null controllability results for the associated linear system and on global Carleman estimates.
Reviewer: Sebastian Anita (Iaşi)
MSC:
| 93B05 | Controllability |
| 35K50 | Systems of parabolic equations, boundary value problems (MSC2000) |
| 35K57 | Reaction-diffusion equations |
References:
| [1] | S. Anita and V. Barbu , Null controllability of nonlinear convective heat equation . ESAIM: COCV 5 ( 2000 ) 157 - 173 . Numdam | MR 1744610 | Zbl 0938.93008 · Zbl 0938.93008 · doi:10.1051/cocv:2000105 |
| [2] | D.G. Aronson and J. Serrin , Local behavior of solutions of quasilinear parabolic equations . Arch. Rational Mech. Anal. 25 ( 1967 ) 81 - 122 . MR 244638 | Zbl 0154.12001 · Zbl 0154.12001 · doi:10.1007/BF00281291 |
| [3] | D.G. Aronson and J. Serrin , A maximum principle for nonlinear parabolic equations . Ann. Scuola Norm. Sup. Pisa 3 ( 1967 ) 291 - 305 Numdam | MR 219901 | Zbl 0148.34803 · Zbl 0148.34803 |
| [4] | J.P. Aubin , L’analyse non linéaire et ses motivations économiques . Masson ( 1984 ). Zbl 0551.90001 · Zbl 0551.90001 |
| [5] | V. Barbu , Exact controllability of the superlinear heat equation . Appl. Math. Optim. 42 ( 2000 ) 73 - 89 . MR 1751309 | Zbl 0964.93046 · Zbl 0964.93046 · doi:10.1007/s002450010004 |
| [6] | T. Cazenave and A. Haraux , Introduction aux problèmes d’évolution semi-linéaires . Ellipses, Paris, Mathématiques & Applications ( 1990 ). Zbl 0786.35070 · Zbl 0786.35070 |
| [7] | S. Cox and E. Zuazua , The rate at which energy decays in a string damped at one end . Indiana Univ. Math. J. 44 ( 1995 ) 545 - 573 . MR 1355412 | Zbl 0847.35078 · Zbl 0847.35078 · doi:10.1512/iumj.1995.44.2001 |
| [8] | A. Doubova , E. Fernández-Cara , M. González-Burgos and E. Zuazua , On the controllability of parabolic system with a nonlinear term involving the state and the gradient . SIAM: SICON (to appear). Zbl 1038.93041 · Zbl 1038.93041 · doi:10.1137/S0363012901386465 |
| [9] | C. Fabre , J.-P. Puel and E. Zuazua , (a) Approximate controllability for the semilinear heat equation. C. R. Acad. Sci. Paris Sér. I Math. 315 (1992) 807-812; (b) Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh Sect. A 125 (1995) 31-61. MR 1318622 | Zbl 0818.93032 · Zbl 0818.93032 · doi:10.1017/S0308210500030742 |
| [10] | C. Fabre , J.-P. Puel and E. Zuazua , Approximate controllability for the linear heat equation with controls of minimal \(L^\infty \) norm . C. R. Acad. Sci. Paris Sér. I Math. 316 ( 1993 ) 679 - 684 . MR 1214415 | Zbl 0799.35094 · Zbl 0799.35094 |
| [11] | E. Fernández-Cara , Null controllability of the semilinear heat equation . ESAIM: COCV 2 ( 1997 ) 87 - 107 . Numdam | Zbl 0897.93011 · Zbl 0897.93011 · doi:10.1051/cocv:1997104 |
| [12] | E. Fernández-Cara and E. Zuazua , The cost of approximate controllability for heat equations: The linear case . Adv. Differential Equations 5 ( 2000 ) 465 - 514 . Zbl 1007.93034 · Zbl 1007.93034 |
| [13] | E. Fernández-Cara and E. Zuazua , Null and approximate controllability for weakly blowing up semilinear heat equations . Ann. Inst. H. Poincaré Anal. Non Linéaire 17 ( 2000 ) 583 - 616 . Numdam | Zbl 0970.93023 · Zbl 0970.93023 · doi:10.1016/S0294-1449(00)00117-7 |
| [14] | E. Fernández-Cara and E. Zuazua , On the null controllability of the one-dimensional heat equation with BV coefficients (to appear). Zbl 1119.93311 · Zbl 1119.93311 |
| [15] | A. Fursikov and O.Yu. Imanuvilov , Controllability of Evolution Equations . Seoul National University, Korea, Lecture Notes 34 ( 1996 ). MR 1406566 | Zbl 0862.49004 · Zbl 0862.49004 |
| [16] | O.Yu. Imanuvilov , Controllability of parabolic equations . Mat. Sb. 186 ( 1995 ) 102 - 132 . MR 1349016 | Zbl 0845.35040 · Zbl 0845.35040 · doi:10.1070/SM1995v186n06ABEH000047 |
| [17] | O.Yu. Imanuvilov and M. Yamamoto , Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications . Lecture Notes in Pure Appl. Math. 218 ( 2001 ) 113 - 137 . MR 1817179 | Zbl 0977.93041 · Zbl 0977.93041 |
| [18] | O.A. Ladyzenskaya , V.A. Solonnikov and N.N. Uraltzeva , Linear and Quasilinear Equations of Parabolic Type . Nauka, Moskow ( 1967 ). Zbl 0174.15403 · Zbl 0174.15403 |
| [19] | A. Pazy , Semigroups of linear operators and applications to partial differential equations . Springer-Verlag, New York ( 1983 ). MR 710486 | Zbl 0516.47023 · Zbl 0516.47023 |
| [20] | D.L. Russell , A unified boundary controllability theory for hyperbolic and parabolic partial differential equations . Stud. Appl. Math. 52 ( 1973 ) 189 - 211 . MR 341256 | Zbl 0274.35041 · Zbl 0274.35041 |
| [21] | F.B. Weissler , Local existence and nonexistence for semilinear parabolic equations in \(L^p\) . Indiana Univ. Math. J. 29 ( 1980 ) 79 - 102 . MR 554819 | Zbl 0443.35034 · Zbl 0443.35034 · doi:10.1512/iumj.1980.29.29007 |
| [22] | F.B. Weissler , Semilinear evolution equations in Banach spaces . J. Funct. Anal. 32 ( 1979 ) 277 - 296 . MR 538855 | Zbl 0419.47031 · Zbl 0419.47031 · doi:10.1016/0022-1236(79)90040-5 |
| [23] | E. Zuazua , Exact boundary controllability for the semilinear wave equation , in Nonlinear Partial Differential Equations and their Applications, Vol. X, edited by H. Brezis and J.-L. Lions. Pitman ( 1991 ) 357 - 391 . MR 1131832 | Zbl 0731.93011 · Zbl 0731.93011 |
| [24] | E. Zuazua , Finite dimensional controllability for the semilinear heat equations . J. Math. Pures 76 ( 1997 ) 570 - 594 . MR 1441986 | Zbl 0872.93014 · Zbl 0872.93014 · doi:10.1016/S0021-7824(97)89951-5 |
| [25] | E. Zuazua , Approximate controllability for semilinear heat equations with globally Lipschitz nonlinearities . Control and Cybernetics 28 ( 1999 ) 665 - 683 . MR 1782020 | Zbl 0959.93025 · Zbl 0959.93025 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
