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By Dan Rafter,J. D. Bogle

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39) lim sup Pn x = 0. n→∞ x∈C Proof If C is precompact then the Hausdorff compactness criterion implies that, for ε -net {x ( j) }, j = 1, 2, . . , m(ε) [25]. e. Pn x = Pn (x −x ( j) )+ Pn x ( j) ≤ Pn ε ε + Pn x ( j) < + Pn x ( j) . 40) 2M 2 We now show that, for sufficiently large n, the inequality Pn x ( j) < ε2 holds for each j = 1, 2, . . , m(ε). In fact, since {ei } is a basis, each element of the net is represented by a convergent series: x ( j) = ∞ i=1 According to the definition of Pn , ( j) ci ei .

The infinite matrix of the operator R(x) and the b column are also of the block form: R(x) = diag(O2×2 , R11 (x), . . , Rk1 (x), R12 (x), . . , Rk2 , . ), 0 Rin (x) = b = (0, 1, 0, − ω2 √ ci λin 0 0 , J11 J21 Jk1 J12 , 0, − , . . , 0, − , 0, − , . )T . 23), the standard multiplication of a matrix and a column is used. Since Jin /ρi are the Fourier coefficients of φ(x) = x + d ∈ L 2 (0, li ) with respect to the basis {u in (x)}∞ n=1 , then ∞ 2 Jin <∞ n=1 for all 1 ≤ i ≤ k, so that b ∈ 2. e. 2 2 2 such ∞ k D(A) = {x ∈ of the operator A consists of all x ∈ : 2 2 ci2 λin (ηin + ξin ) < ∞}.

1). , [1, 19, 20] for references and other models). The rigid body can rotate around the fixed point O under the action of a control torque M. We suppose that each beam is attached at the distance d from the point O, and that li is the length of the ith beam. Let wi (x, t) be the deflection of the ith beam from the axis Oi xi at the location x ∈ [0, li ] and time t ≥ 0. Thus, the evolution of the system is defined by the following functions: θ (t), wi (x, t), x ∈ [0, li ], t ≥ 0, i = 1, k, where θ is the angle between the axes O x0 and O x1 .

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