Some uniqueness and observability problems arising in the control of vibrations
Journées équations aux dérivées partielles (1999), article no. 19, 8 p.

We discuss a control problem for the Lamé system which naturally leads to the following uniqueness problem: Given a bounded domain of 𝐑 3 , 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},
     eid = {19},
     pages = {1--8},
     publisher = {Universit\'e de Nantes},
     year = {1999},
     language = {en},
     url = {http://archive.numdam.org/item/JEDP_1999____A19_0/}
}
TY  - JOUR
AU  - Zuazua, Enrique
TI  - Some uniqueness and observability problems arising in the control of vibrations
JO  - Journées équations aux dérivées partielles
PY  - 1999
SP  - 1
EP  - 8
PB  - Université de Nantes
UR  - http://archive.numdam.org/item/JEDP_1999____A19_0/
LA  - en
ID  - JEDP_1999____A19_0
ER  - 
%0 Journal Article
%A Zuazua, Enrique
%T Some uniqueness and observability problems arising in the control of vibrations
%J Journées équations aux dérivées partielles
%D 1999
%P 1-8
%I Université de Nantes
%U http://archive.numdam.org/item/JEDP_1999____A19_0/
%G en
%F JEDP_1999____A19_0
Zuazua, Enrique. Some uniqueness and observability problems arising in the control of vibrations. Journées équations aux dérivées partielles (1999), article  no. 19, 8 p. http://archive.numdam.org/item/JEDP_1999____A19_0/

[BLR] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Cont. Optim., 30 (1992), 1024-1065. | MR | Zbl

[BuL] N. Burq and G. Lebeau, work in preparation.

[LZ] G. Lebeau and E. Zuazua, Decay rates for the linear system of three-dimensional thermoelasticity, Archives Rat. Mech. Anal., to appear (C. R. Acad. Sci. Paris, 324 (1997), 409-415). | MR | Zbl

[LiZ] J.L. Lions and E. Zuazua, A generic uniqueness result for the Stokes system and its control theoretical consequences, in «Partial Differential Equations and Applications», P. Marcellini, G. Talenti and E. Visentini eds., Marcel-Dekker Inc., LNPAS 177, 1996, p. 221-235. | MR | Zbl

[M] A. Mcnabb, Strong comparison theorems for elliptic equations of second order, J. Math. Mech., 10 (1961), 431-440. | MR | Zbl

[PZ] G. Perla-Menzala and E. Zuazua, Energy decay of magnetoelastic waves in a bounded conductive medium, Asymptotic Anal., 18 (1998), 349-362. | MR | Zbl

[R] J. Ralston, Solution of the wave equation with localized energy, Comm. Pure Appl. Math., 22 (1969), 807-823. | MR | Zbl

[SZ] G. Sweers and E. Zuazua, On the nonexistence of some special eigenfunctions for the Dirichlet Laplacian and the Lamé system, J. Elasticity, 52 (1999), 111-120. | MR | Zbl

[Z] E. Zuazua, A uniqueness result for the linear system of elasticity and its control theoretical consequences, SIAM J. Cont. Optim., 34 (5) (1996), 1473-1495 & 37 (1) (1998), 330-331. | MR | Zbl