Appl Math Optim (2009) 60: 71–103
On Controllability of an Elastic Ring
Sergei A. Avdonin · Boris P. Belinskiy ·
Sergei A. Ivanov
Published online: 7 February 2009
© Springer Science+Business Media, LLC 2009
Abstract We study the exact controllability problem for a ring under stretching ten-
sion that varies in time. We are looking for a couple of forces, which drive the state
solution to rest. We show that applying two forces is necessary for controllability
and the ring is controllable in the time interval greater than the optical length of the
string. We also explain why one force would not be enough to control the ring. We
use the method of moments to reduce the controllability problem to a moment prob-
lem for the controlling forces. The solution of that problem is based on an auxiliary
basis property result. Both method of moments and proof of the basis property are
developed for the model with time-dependent parameters.
Keywords Distributed parameter systems · Wave equation · Control · Bases
of functions · Sobolev spaces
Communicated by Irena Lasiecka.
S.A.’s research was supported in part by the NSF, grant ARC–0724860.
B.B.’s research was supported in part by University of Tennessee at Chattanooga Faculty Research
S.I.’s research was supported in part by the Russian Foundation for Basic Research, grants
08-01-00595a and 08-01-00676a.
Department of Mathematical Sciences, University of Alaska, Fairbanks, AK 99775-6660, USA
B.P. Belinskiy (
Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403-2598,
Institute of Laser Physics, St. Petersburg State University, St. Petersburg, 198904, Russia