Contemporary Mathematics Volume 209, 1997 BOUNDARY OBSERVABILITY AND CONTROLLABILITY OF LINEAR ELASTODYNAMIC SYSTEMS FATIHA ALA8AU AND VILMOS KOMORNIK ABSTRACT. In [10) Lions obtained several boundary observability and exact controllability results for homogeneous and isotropic linear elastodynamic sys- tems. We generalize these results to the non-isotropic case. This requires some new identities. We also study the optimality of our estimates. 1. 08SERVABILITY Let (aijkt) be a tensor such that aijkl = a ikl = aklij (all indices run over the integers 1, 2, 3), satisfying for some a 0 the ellipticity condition (1) for every symmetric tensor €ij. (Here and in the sequel we shall use the summation convention for repeated indices.) Let {} be a non-empty bounded open set in JR3 having a boundary r of class C2• Given a function e = (e 17 e 2 , 6) : 0 -t IR3 , we shall use the notations €ij = ~(ei,j + e ,i), O'ij = aijkl€kl, where ei,i = aeif8x and e ,i = ae /8xi. Hit is necessary to be more precise, we shall write ei (e) and ai (e) instead of ei , aii· Consider the problem { e~'- O'ij,j = 0 in {} X IR, ei = 0 on r X IR, ~i(o) = e? and eHo) = el in n, ~ = 1,2,3. (2) We recall (see e.g. [2] or [4]) that this problem is well-posed in the following sense: (i) Given (eo,el) E HJ(0)3 x L2(0)3 arbitrarily, the problem (2) has a unique (so-called weak) solution e E C(IR HJ(0)3 ) n C1(1R L 2 (0)3 ) 1991 Mathematics Subject Classification. 35L55, 35Q72, 93805, 93807, 93C20. Key words and phrases. observability, controllability, partial differential equation, elasticity. The second author is grateful to the organisers of the conference for their invitation and to the INRIA Lorraine (Projet Numath) for supporting his travel expenses. © 1997 American Mathematical Society 1
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