Article
MSC:
30E05,
35L35,
35Q74,
58J45,
74K20,
93B05 | MR 3304785 | Zbl 06433701 | DOI: 10.1007/s10587-014-0140-7
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
beam equation; null-controllability; structural damping; moment problem; biorthogonals
beam equation; null-controllability; structural damping; moment problem; biorthogonals
Summary:
This paper is devoted to studying the effects of a vanishing structural damping on the controllability properties of the one dimensional linear beam equation. The vanishing term depends on a small parameter $\varepsilon \in (0,1)$. We study the boundary controllability properties of this perturbed equation and the behavior of its boundary controls $v_{\varepsilon }$ as $\varepsilon $ goes to zero. It is shown that for any time $T$ sufficiently large but independent of $\varepsilon $ and for each initial data in a suitable space there exists a uniformly bounded family of controls $(v_\varepsilon )_\varepsilon $ in $L^2(0, T)$ acting on the extremity $x = \pi $. Any weak limit of this family is a control for the beam equation. This analysis is based on Fourier expansion and explicit construction and evaluation of biorthogonal sequences. This method allows us to measure the magnitude of the control needed for each eigenfrequency and to show their uniform boundedness when the structural damping tends to zero.
References:
[1] Avdonin, S. A., Ivanov, S. A.: Families of Exponentials. The method of moments in controllability problems for distributed parameter systems. Cambridge University Press Cambridge (1995). MR 1366650 | Zbl 0866.93001
[2] Ball, J. M., Slemrod, M.: Nonharmonic Fourier series and the stabilization of distributed semilinear control systems. Commun. Pure Appl. Math. 32 (1979), 555-587. DOI 10.1002/cpa.3160320405 | MR 0528632
[3] DiPerna, R. J.: Convergence of approximate solutions to conservation laws. Arch. Ration. Mech. Anal. 82 (1983), 27-70. DOI 10.1007/BF00251724 | MR 0684413 | Zbl 0519.35054
[4] Edward, J.: Ingham-type inequalities for complex frequencies and applications to control theory. J. Math. Anal. Appl. 324 (2006), 941-954. DOI 10.1016/j.jmaa.2005.12.074 | MR 2265092 | Zbl 1108.26019
[5] Edward, J., Tebou, L.: Internal null-controllability for a structurally damped beam equation. Asymptotic Anal. 47 (2006), 55-83. MR 2224406 | Zbl 1098.35096
[6] Fattorini, H. O., Russell, D. L.: Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations. Q. Appl. Math. 32 (1974), 45-69. DOI 10.1090/qam/510972 | MR 0510972 | Zbl 0281.35009
[7] Fattorini, H. O., Russell, D. L.: Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Ration. Mech. Anal. 43 (1971), 272-292. DOI 10.1007/BF00250466 | MR 0335014 | Zbl 0231.93003
[8] Ignat, L. I., Zuazua, E.: Dispersive properties of numerical schemes for nonlinear Schrödinger equations. Foundations of Computational Mathematics; Santander, Spain, 2005, London Math. Soc. Lecture Note Ser. 331 Cambridge University Press, Cambridge (2006), 181-207 L. M. Pardo et al. MR 2277106 | Zbl 1106.65321
[9] Ignat, L. I., Zuazua, E.: Dispersive properties of a viscous numerical scheme for the Schrödinger equation. C. R., Math., Acad. Sci. Paris 340 (2005), 529-534. DOI 10.1016/j.crma.2005.02.024 | MR 2135236 | Zbl 1063.35016
[10] Ingham, A. E.: Some trigonometrical inequalities with applications to the theory of series. Math. Z. 41 (1936), 367-379. DOI 10.1007/BF01180426 | MR 1545625 | Zbl 0014.21503
[11] Komornik, V., Loreti, P.: Fourier Series in Control Theory. Springer Monographs in Mathematics Springer, New York (2005). MR 2114325 | Zbl 1094.49002
[12] Lebeau, G., Zuazua, E.: Null-controllability of a system of linear thermoelasticity. Arch. Ration. Mech. Anal. 141 (1998), 297-329. DOI 10.1007/s002050050078 | MR 1620510 | Zbl 1064.93501
[13] Lions, J.-L.: Exact controllability, perturbations and stabilization of distributed systems. Volume 1: Exact controllability. Research in Applied Mathematics 8 Masson, Paris French (1988). MR 0963060
[14] Micu, S., RovenĹŁa, I.: Uniform controllability of the linear one dimensional Schrödinger equation with vanishing viscosity. ESAIM, Control Optim. Calc. Var. 18 (2012), 277-293. DOI 10.1051/cocv/2010055 | MR 2887936
[15] Seidman, T. I., Avdonin, S. A., Ivanov, S. A.: The `window problem' for series of complex exponentials. J. Fourier Anal. Appl. 6 (2000), 233-254. DOI 10.1007/BF02511154 | MR 1755142 | Zbl 0960.42012
[16] Young, R. M.: An Introduction to Nonharmonic Fourier Series. Pure and Applied Mathematics 93 Academic Press, New York (1980). MR 0591684 | Zbl 0493.42001
[17] Zabczyk, J.: Mathematical Control Theory: An Introduction. Systems & Control: Foundations & Applications Birkhäuser, Boston (1992). MR 1193920 | Zbl 1071.93500
Search
Partner of
