Insensitizing controls for a class of nonlinear Ginzburg-Landau equations. (English) Zbl 1307.93074
Summary: This paper shows the existence of insensitizing controls for a class of nonlinear complex Ginzburg-Landau equations with homogeneous Dirichlet boundary conditions and arbitrarily located internal controller. When the nonlinearity in the equation satisfies a suitable superlinear growth condition at infinity, the existence of insensitizing controls for the corresponding semilinear Ginzburg-Landau equation is proved. Meanwhile, if the nonlinearity in the equation is only a smooth function without any additional growth condition, a local result on insensitizing controls is obtained. As usual, the problem of insensitizing controls is transformed into a suitable controllability problem for a coupled system governed by a semilinear complex Ginzburg-Landau equation and a linear one through one control. The key is to establish an observability inequality for a coupled linear Ginzburg-Landau system with one observer.
MSC:
| 93B05 | Controllability |
| 93B07 | Observability |
| 93C20 | Control/observation systems governed by partial differential equations |
| 93C10 | Nonlinear systems in control theory |
| 35Q56 | Ginzburg-Landau equations |
References:
| [1] | Ammar-Khodja F, Benabdallah A, González-Burgos M, et al. Recent results on the controllability of linear coupled parabolic problems: A survey. Math Control Relat Fields, 2011, 1: 267-306 · Zbl 1235.93041 · doi:10.3934/mcrf.2011.1.267 |
| [2] | Bodart O, González-Burgos M, Pérez-Garcia R. A local result on insensitizing controls for a semilinear heat equation with nonlinear boundary Fourier conditions. SIAM J Control Optim, 2004, 43: 955-969 · Zbl 1101.93009 · doi:10.1137/S036301290343161X |
| [3] | Bodart O, Gronzález-Burgos M, Pérez-Garcia R. Existence of insensitizing controls for a semilinear heat equation with a superlinear nonlinearity. Comm Partial Differential Equations, 2004, 29: 1017-1050 · Zbl 1067.93035 · doi:10.1081/PDE-200033749 |
| [4] | Carreño N. Local controllability of the N-dimensional Boussinesq system with N − 1 scalar controls in an arbitrary control domain. Math Control Relat Fields, 2012, 2: 361-382 · Zbl 1259.35214 · doi:10.3934/mcrf.2012.2.361 |
| [5] | Coron J-M. Control and Nonlinearity, Mathematical Surveys and Monographs, vol. 136. Providence, RI: Amer Math Soc, 2007 · Zbl 1140.93002 |
| [6] | Fu X. Null controllability for the parabolic equation with a complex principal part. J Funct Anal, 2009, 257: 1333-1354 · Zbl 1178.35099 · doi:10.1016/j.jfa.2009.05.024 |
| [7] | Fursikov A V, Yu O. Imanuvilov, Controllability of Evolution Equations. Lecture Notes Series 34. Seoul: Seoul National University, 1996 · Zbl 0862.49004 |
| [8] | Ladyženskaja O A. Boundary Value Problems of Mathematical Physics, III. Proceedings of the Steklov Institute of Mathematics, no. 83. Providence, RI: Amer Math Soc, 1965 · Zbl 0164.12501 |
| [9] | Lions, J-L, Quelques notions dans l’analyse et le contrôle de systèmes à données incomplètes, 43-54 (1990), Málaga |
| [10] | Liu X. Insensitizing controls for a class of quasilinear parabolic equations. J Differential Equations, 2012, 253: 1287-1316 · Zbl 1251.35039 · doi:10.1016/j.jde.2012.05.018 |
| [11] | Liu X. Global Carleman estimate for stochastic parabolic equations, and its application. ESAIM Control Optim Calc Var, 2014, 20: 823-839 · Zbl 1292.93032 · doi:10.1051/cocv/2013085 |
| [12] | Lü Q. Some results on the controllability of forward stochastic heat equations with control on the drift. J Funct Anal, 2011, 260: 832-851 · Zbl 1213.60105 · doi:10.1016/j.jfa.2010.10.018 |
| [13] | Rosier L, Zhang B. Null controllability of the complex Ginzburg-Landau equation. Ann Inst H Poincaré Anal Non Linéaire, 2009, 26: 649-673 · Zbl 1170.35095 · doi:10.1016/j.anihpc.2008.03.003 |
| [14] | de Teresa L. Insensitizing controls for a semilinear heat equation. Comm Partial Differential Equations, 2000, 25: 39-72 · Zbl 0942.35028 · doi:10.1080/03605300008821507 |
| [15] | Wang G. L∞-null controllability for the heat equation and its consequences for the time optimal control problem. SIAM J Control Optim, 2008, 47: 1701-1720 · Zbl 1165.93016 · doi:10.1137/060678191 |
| [16] | Yan Y, Sun F. Insensitizing controls for a forward stochastic heat equation. J Math Anal Appl, 2011, 384: 138-150 · Zbl 1226.60093 · doi:10.1016/j.jmaa.2011.05.058 |
| [17] | Zhang, X., A unified controllability/observability theory for some stochastic and deterministic partial differential euqations, 3008-3034 (2010), India · Zbl 1226.93031 |
| [18] | Zuazua, E., Controllability and observability of partial differential equations: Some results and open problems, 527-621 (2006), Amsterdam · Zbl 1193.35234 |
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