We consider the problem of boundary control of an elastic system with coupling to a potential equation. The potential equation represents the linearized motions of an incompressible inviscid fluid in a cavity bounded in part by an elastic membrane. Sufficient control is placed on a portion of the elastic membrane to insure that the uncoupled membrane is exactly controllable. The main result is that if the density of the fluid is sufficiently small, then the coupled system is exactly controllable.
@article{bwmeta1.element.bwnjournal-article-amcv11i6p1231bwm,
author = {Hansen, Scott},
title = {Exact controllability of an elastic membrane coupled with a potential fluid},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {11},
year = {2001},
pages = {1231-1248},
zbl = {1005.93024},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv11i6p1231bwm}
}
Hansen, Scott. Exact controllability of an elastic membrane coupled with a potential fluid. International Journal of Applied Mathematics and Computer Science, Tome 11 (2001) pp. 1231-1248. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv11i6p1231bwm/
[000] Avalos G. (1996): The exponential stability of coupled hyperbolic/parabolic system arising in structural acoustics. — Abstr. Appl. Anal., Vol.1, No.2, pp.203–219. | Zbl 0989.93078
[001] Banks H.T., Silcox R.J. and Smith R.C. (1993): The modeling and control of acoustic/structure interaction problems via peizoceramic actuators: 2-d numerical examples. — ASME J. Vibr. Acoust., Vol.2, pp.343–390.
[002] Bardos C., Lebeau G. and Rauch J. (1992): Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. — SIAM J. Contr., Vol.30, No.5, pp.1024–1065. | Zbl 0786.93009
[003] Conca C., Osses A. and Planchard J. (1998): Asympotic analysis relating spectral models in fluid-solid vibrations. — SIAM J. Num. Anal., Vol.35, No.3, pp.1020–1048. | Zbl 0911.35080
[004] Galdi G.D., (1994): An Introduction to the Mathematical Theory of Navier-Stokes Equations, Vol.I. — New York: Springer. | Zbl 0949.35004
[005] Hansen S.W. and Lyashenko A. (1997): Exact controllability of a beam in an incompressible inviscid fluid. — Disc. Cont. Dyn. Syst., Vol.3., No.1, pp.59–78. | Zbl 0969.74561
[006] Lighthill J. (1981):, Energy flow in the cochlea. — J. Fluid Mech., Vol.106, pp.149–213. | Zbl 0474.76074
[007] Lions J.-L. (1988): Exact controllability, stabilization and perturbations for distributed systems. — SIAM Rev. Vol.30, No.1, pp.1–67. | Zbl 0644.49028
[008] Lions J.-L. and Zuazua E. (1995): Approximate controllability of hydro-elastic coupled system. — ESAIM: Contr. Optim. Calc. Var., Vol.1, pp.1–15. | Zbl 0878.93034
[009] Micu S. and Zuazua E. (1997): Boundary controllability of a linear hybrid system arising in the control of noise. — SIAM J. Contr. Optim., Vol.35, No.5, pp.1614–1637. | Zbl 0888.35017
[010] Necas J. (1967): Les Méthodes directes en théorie des équations elliptiques. — Paris: Masson.
[011] Osses A. and Puel J.-P. (1998): Boundary controllability of a stationary stokes system with linear convection observed on an interior curve. — J. Optim. Theory Appl., Vol.99, No.1, pp.201–234. | Zbl 0958.93010
[012] Osses A. and Puel J.-P. (1999): Approximate controllability for a linear model of fluid structure interaction. — ESAIM Contr. Optim. Calc. Var., Vol.4, No.??, pp.497–519. | Zbl 0931.35014
[013] Pazy A. (1983): Semigroups of Linear Operators and Applications and Partial Differential Equations. — New York: Springer-Verlag. | Zbl 0516.47023