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Abstract. In this work we study a transmission problem for the model of beams developed by S.P. Timoshenko [10]. We consider the case of mixed material, that is, a part of the beam has friction and the other is purely elastic. We show that for this type of material, the dissipation produced by the frictional part is strong enough to produce exponential decay of the solution, no matter how small is its size. We use the method of energy to prove exponential decay for the solution.Mathematical subject classification: 35J55, 35J77, 93C20.

In this paper we study exact boundary controllability for a system of two linear wave equations coupled by lower order terms. We obtain square integrable control of Neuman type for initial state with finite energy, in nonsmooth domains of the plane. PreliminariesLet Ω ⊂ R 2 be a bounded simply connected domain with piecewise C ∞ boundary ∂Ω. It is assumed that ∂Ω has no cuspid points and that Ω lays in one side of ∂Ω. We will refer to this type of domain as curvilinear polygon. The exterior unit normal vector η is defined almost everywhere in ∂Ω. Points of R 2 × R are denoted by (x 1 , x 2 , t), denotes the Laplacian operator in the spatial variable x = (x 1 , x 2 ) ∈ R 2 and subscripts are used for partial derivatives. Consider the systemwith T > 0, α, β, γ , δ real constants and ∂Ω = Γ 0 ∪ Γ 1 is a decomposition of the boundary where the parts Γ 0 and Γ 1 have disjoint interiors but their closure may have common isolated points. Many problems in structural dynamics are modeled by this type of system. For instance, the equations (1.2) describe the transverse free vibrations of an elastically connected double-membrane compound system. In this case u and v are the displacements of two vibrating membranes measured from their equilibrium positions. The coupling terms ±α(u − v) represent forces imposed by the coupling elastic layer. If the membranes are held fixed on the portion Γ 0 of the * Corresponding author.

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We establish exact boundary controllability for the wave equation in a polyhedral domain where a part of the boundary moves slowly with constant speed in a small interval of time. The control on the moving part of the boundary is given by the conormal derivative associated with the wave operator while in the fixed part the control is of Neuman type. For initial state H I x L 2 we obtain controls in L 2. ~)