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Approximate controllability of nonsimple elastic plate with memory

  • * Corresponding author: Moncef Aouadi

    * Corresponding author: Moncef Aouadi 
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  • In this paper, we give some qualitative results on the behavior of a nonsimple elastic plate with memory corresponding to anti-plane shear deformations. First we describe briefly the equations of the considered plate and then we study the well-posedness of the resulting problem. Secondly, we perform the spectral analysis, in particular, we establish conditions on the physical constants of the plate to guarantee the simplicity and the monotonicity of the roots of the characteristic equation. The spectral results are used to prove the exponential stability of the solutions at an optimal decay rate given by the physical constants. Then we present an approximate controllability result of the considered control problem. Finally, we give some numerical experiments based on the spectral method developed with implementation in MATLAB for one and two-dimensional problems.

    Mathematics Subject Classification: Primary: 93B05; Secondary: 35B35.


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  • Figure 1.  The displacement $ u(x,t) $ for time interval $ [0,5] $ (left) on the points $ x_0 = 1/6, x_1 = 3/4 $ and $ x_2 = 1 $ (right)

    Figure 2.  The Energy $ E(t) $ in the time interval $ [0, 5] $ (left) the logarithmic energy (right)

    Figure 3.  The displacement $ u(x_i,y_i,t) $ at $ (x_i, y_i) = (1,1) $, $ (\frac{1}{2},\frac{3}{4}) $ and $ (\frac{1}{3},\frac{5}{6}) $, for time $ [0,20] $

    Figure 4.  The energy $ E(t) $ in the time interval $ [0, 20] $ (left) the logarithmic energy (right)

    Figure 5.  The eigenvalues $ n\mapsto \sigma_{0}(n) $ and $ n\mapsto\Re e(\sigma_{1})(n) $

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