zbMATH — the first resource for mathematics

A uniqueness result for the linear system of elasticity and its control theoretical consequences. (English) Zbl 0856.73010
SIAM J. Control Optimization 34, No. 5, 1473-1495 (1996); erratum ibid. 37, No. 1, 330-331 (1998).
We consider the time-dependent linear system of three-dimensional elasticity for an homogeneous and isotropic elastic body \(\Omega\) with Dirichlet boundary conditions. We prove that when the domain \(\Omega\) is of class \(C^2\) and is not symmetric with respect to a plane parallel to \(x_3=0\), any solution whose first two components vanish in some open subset of \(\Omega\) for a large enough time interval has to be identically zero. As a consequence of this uniqueness result, we prove that the linear system of elasticity is approximately controllable by means of planar volume forces with zero third component. We show that these results fail for some Lipschitz domains that are piecewise smooth and symmetric with respect to the plane \(x_3=0\).

74G30 Uniqueness of solutions of equilibrium problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
74B05 Classical linear elasticity
93C20 Control/observation systems governed by partial differential equations
Full Text: DOI