We discuss a control problem for the Lamé system which naturally leads to the following uniqueness problem: Given a bounded domain of , are there non-trivial solutions of the evolution Lamé system with homogeneous Dirichlet boundary conditions for which the first two components vanish? We show that such solutions do not exist when the domain is Lipschitz. However, in two space dimensions one can build easily polygonal domains in which there are eigenvibrations with the first component being identically zero. These uniqueness problems do not feet in the context of the classical Cauchy problem. They are of global nature and, therefore, the geometry of the domain under consideration plays a key role. We also present a list of related open problems.
@article{JEDP_1999____A19_0,
author = {Zuazua, Enrique},
title = {Some uniqueness and observability problems arising in the control of vibrations},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
year = {1999},
pages = {1-8},
language = {en},
url = {http://dml.mathdoc.fr/item/JEDP_1999____A19_0}
}
Zuazua, Enrique. Some uniqueness and observability problems arising in the control of vibrations. Journées équations aux dérivées partielles, (1999), pp. 1-8. http://gdmltest.u-ga.fr/item/JEDP_1999____A19_0/
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