# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2021147
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## Approximate controllability of nonsimple elastic plate with memory

 1 Université de Carthage, Ecole Nationale d'Ingénieurs de Bizerte, 7035, BP 66, Tunisia 2 UR Systèmes dynamiques et applications, UR 17ES21 3 Université de Monastir, Faculté des Sciences de Monastir, 5000-Monastir, Tunisia 4 South Ural State University, Lenin prospect 76, Chelyabinsk, 454080, Russian Federation 5 Université de Monastir, Ecole Nationale d'Ingénieurs de Monastir, 5005, Tunisia

Received  July 2021 Revised  September 2021 Early access November 2021

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.

Citation: Moncef Aouadi, Imed Mahfoudhi, Taoufik Moulahi. Approximate controllability of nonsimple elastic plate with memory. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021147
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##### References:
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)
The Energy $E(t)$ in the time interval $[0, 5]$ (left) the logarithmic energy (right)
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 $[0,20]$
The energy $E(t)$ in the time interval $[0, 20]$ (left) the logarithmic energy (right)
The eigenvalues $n\mapsto \sigma_{0}(n)$ and $n\mapsto\Re e(\sigma_{1})(n)$
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