# American Institute of Mathematical Sciences

May  2022, 15(5): 1015-1043. doi: 10.3934/dcdss.2021147

## 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

* Corresponding author: Moncef Aouadi

Received  July 2021 Revised  September 2021 Published  May 2022 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 and Continuous Dynamical Systems - S, 2022, 15 (5) : 1015-1043. doi: 10.3934/dcdss.2021147
##### References:
 [1] A. Anikushyn and M. Pokojovy, Multidimensional thermoelasticity for nonsimple materials-well-posedness and long-time behavior, Appl. Anal., 96 (2017), 1561-1585.  doi: 10.1080/00036811.2017.1295447. [2] M. Aouadi, Stability aspects in a nonsimple thermoelastic diffusion problem, Appl. Anal., 92 (2013), 1816-1828.  doi: 10.1080/00036811.2012.702341. [3] M. Aouadi, On uniform decay of a nonsimple thermoelastic bar with memory, J. Math. Anal. Appl., 402 (2013), 745-757.  doi: 10.1016/j.jmaa.2013.01.059. [4] M. Aouadi and T. Moulahi, The controllability of a thermoelastic plate problem revisited, Evol. Equ. Control Theory, 7 (2018), 1-31.  doi: 10.3934/eect.2018001. [5] M. Aouadi and T. Moulahi, Asymptotic analysis of a nonsimple thermoelastic rod, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1475-1492.  doi: 10.3934/dcdss.2016059. [6] M. Aouadi and T. Moulahi, Approximate controllability of abstract nonsimple thermoelastic problem, Evol. Equ. Control Theory, 4 (2015), 373-389.  doi: 10.3934/eect.2015.4.373. [7] S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, Graduate Texts in Mathematics, 137. Springer-Verlag, New York, 1992. doi: 10.1007/b97238. [8] M. Bachher and N. Sarkar, Nonlocal theory of thermoelastic materials with voids and fractional derivative heat transfer,, Waves Random Complex Media, 29 (2019), 595-613.  doi: 10.1080/17455030.2018.1457230. [9] S. Biswas, The propagation of plane waves in nonlocal visco-thermoelastic porous medium based on nonlocal strain gradient theory,, Waves in Random and Complex Media, 2021. doi: 10.1080/17455030.2021.1909780. [10] M. Ciarletta and D. Ieşan, Non-classical Elastic Solids, Pitman Research Notes in Mathematical Series, vol. 293, John Wiley & Sons, Inc., New York, 1993. [11] R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems, Lecture Notes in Control and Information Sciences, Springer, Berlin, 1978. [12] R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, 21, Springer, New York, NY, USA, 1995. doi: 10.1007/978-1-4612-4224-6. [13] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609. [14] C. A. J. Fletcher, Computational Galerkin Methods, Springer Series in Computational Physics, Springer-Verlag, New York, 1984. doi: 10.1007/978-3-642-85949-6. [15] C. Giorgi, A. Marzocchi and V. Pata, Asymptotic behavior of a semilinear problem in heat conduction with memory, NoDEA Nonlinear Diff. Equat. Appl., 5 (1998), 333-354.  doi: 10.1007/s000300050049. [16] D. Y. Khusainov and M. Pokojovy, Solving the linear 1D thermoelasticity equations with pure delay, Int. J. Math. Math. Sci., 2015 (2015), Art. ID 479267, 11 pp. [17] G. Leugering, Time optimal boundary controllability of a simple linear viscoelastic liquid, Math. Methods Appl. Sci., 9 (1987), 413-430. [18] Z. Liu, A. Magaña and R. Quintanilla, On the time decay of solutions for non-simple elasticity with voids, Z. Angew. Math. Mech., 96 (2016), 857-873.  doi: 10.1002/zamm.201400290. [19] P. Loreti, L. Pandolfi and D. Sforza, Boundary controllability and observability of a viscoelastic string, SIAM J. Control Optim., 50 (2012), 820-844.  doi: 10.1137/110827740. [20] P. Loreti and D. Sforza, Reachability problems for a class of integro-differential equations, J. Differential Equations, 248 (2010), 1711-1755.  doi: 10.1016/j.jde.2009.09.016. [21] Q. Lü, X. Zhang and E. Zuazua, Null controllability for wave equations with memory, J. Math. Pures Appl., 108 (2017), 500-531.  doi: 10.1016/j.matpur.2017.05.001. [22] A. Magaña and R. Quintanilla, Exponential decay in nonsimple thermoelasticity of type III, Math. Meth. Appl. Sci., 39 (2016), 225-235.  doi: 10.1002/mma.3472. [23] R. D. Mindlin, Micro-structure in linear elasticity, Arch. Ration. Mech. Anal., 16 (1964), 51-78.  doi: 10.1007/BF00248490. [24] V. Pata and R. Quintanilla, On the decay of solutions in nonsimple elastic solids with memory, J. Math. Anal. Appl., 363 (2010), 19-28.  doi: 10.1016/j.jmaa.2009.07.055. [25] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [26] W. Press, S. Teukolsky, W. Vetterling and B. Flannery, Numerical Recipes in FORTRAN, Cambridge University Press, Cambridge, 1992. [27] M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987. [28] R. A. Toupin, Theories of elasticity with couple-stress, Arch. Ration. Mech. Anal., 17 (1964), 85-112.  doi: 10.1007/BF00253050. [29] L. N. Trefethen, Spectral Methods in Matlab, SIAM Press, volume 10, 2000.  doi: 10.1137/1.9780898719598. [30] J. Zabczyk, Mathematical Control Theory: An Introduction, Reprint of the 1995 edition, Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2008. doi: 10.1007/978-0-8176-4733-9.

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##### References:
 [1] A. Anikushyn and M. Pokojovy, Multidimensional thermoelasticity for nonsimple materials-well-posedness and long-time behavior, Appl. Anal., 96 (2017), 1561-1585.  doi: 10.1080/00036811.2017.1295447. [2] M. Aouadi, Stability aspects in a nonsimple thermoelastic diffusion problem, Appl. Anal., 92 (2013), 1816-1828.  doi: 10.1080/00036811.2012.702341. [3] M. Aouadi, On uniform decay of a nonsimple thermoelastic bar with memory, J. Math. Anal. Appl., 402 (2013), 745-757.  doi: 10.1016/j.jmaa.2013.01.059. [4] M. Aouadi and T. Moulahi, The controllability of a thermoelastic plate problem revisited, Evol. Equ. Control Theory, 7 (2018), 1-31.  doi: 10.3934/eect.2018001. [5] M. Aouadi and T. Moulahi, Asymptotic analysis of a nonsimple thermoelastic rod, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1475-1492.  doi: 10.3934/dcdss.2016059. [6] M. Aouadi and T. Moulahi, Approximate controllability of abstract nonsimple thermoelastic problem, Evol. Equ. Control Theory, 4 (2015), 373-389.  doi: 10.3934/eect.2015.4.373. [7] S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, Graduate Texts in Mathematics, 137. Springer-Verlag, New York, 1992. doi: 10.1007/b97238. [8] M. Bachher and N. Sarkar, Nonlocal theory of thermoelastic materials with voids and fractional derivative heat transfer,, Waves Random Complex Media, 29 (2019), 595-613.  doi: 10.1080/17455030.2018.1457230. [9] S. Biswas, The propagation of plane waves in nonlocal visco-thermoelastic porous medium based on nonlocal strain gradient theory,, Waves in Random and Complex Media, 2021. doi: 10.1080/17455030.2021.1909780. [10] M. Ciarletta and D. Ieşan, Non-classical Elastic Solids, Pitman Research Notes in Mathematical Series, vol. 293, John Wiley & Sons, Inc., New York, 1993. [11] R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems, Lecture Notes in Control and Information Sciences, Springer, Berlin, 1978. [12] R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, 21, Springer, New York, NY, USA, 1995. doi: 10.1007/978-1-4612-4224-6. [13] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609. [14] C. A. J. Fletcher, Computational Galerkin Methods, Springer Series in Computational Physics, Springer-Verlag, New York, 1984. doi: 10.1007/978-3-642-85949-6. [15] C. Giorgi, A. Marzocchi and V. Pata, Asymptotic behavior of a semilinear problem in heat conduction with memory, NoDEA Nonlinear Diff. Equat. Appl., 5 (1998), 333-354.  doi: 10.1007/s000300050049. [16] D. Y. Khusainov and M. Pokojovy, Solving the linear 1D thermoelasticity equations with pure delay, Int. J. Math. Math. Sci., 2015 (2015), Art. ID 479267, 11 pp. [17] G. Leugering, Time optimal boundary controllability of a simple linear viscoelastic liquid, Math. Methods Appl. Sci., 9 (1987), 413-430. [18] Z. Liu, A. Magaña and R. Quintanilla, On the time decay of solutions for non-simple elasticity with voids, Z. Angew. Math. Mech., 96 (2016), 857-873.  doi: 10.1002/zamm.201400290. [19] P. Loreti, L. Pandolfi and D. Sforza, Boundary controllability and observability of a viscoelastic string, SIAM J. Control Optim., 50 (2012), 820-844.  doi: 10.1137/110827740. [20] P. Loreti and D. Sforza, Reachability problems for a class of integro-differential equations, J. Differential Equations, 248 (2010), 1711-1755.  doi: 10.1016/j.jde.2009.09.016. [21] Q. Lü, X. Zhang and E. Zuazua, Null controllability for wave equations with memory, J. Math. Pures Appl., 108 (2017), 500-531.  doi: 10.1016/j.matpur.2017.05.001. [22] A. Magaña and R. Quintanilla, Exponential decay in nonsimple thermoelasticity of type III, Math. Meth. Appl. Sci., 39 (2016), 225-235.  doi: 10.1002/mma.3472. [23] R. D. Mindlin, Micro-structure in linear elasticity, Arch. Ration. Mech. Anal., 16 (1964), 51-78.  doi: 10.1007/BF00248490. [24] V. Pata and R. Quintanilla, On the decay of solutions in nonsimple elastic solids with memory, J. Math. Anal. Appl., 363 (2010), 19-28.  doi: 10.1016/j.jmaa.2009.07.055. [25] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [26] W. Press, S. Teukolsky, W. Vetterling and B. Flannery, Numerical Recipes in FORTRAN, Cambridge University Press, Cambridge, 1992. [27] M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987. [28] R. A. Toupin, Theories of elasticity with couple-stress, Arch. Ration. Mech. Anal., 17 (1964), 85-112.  doi: 10.1007/BF00253050. [29] L. N. Trefethen, Spectral Methods in Matlab, SIAM Press, volume 10, 2000.  doi: 10.1137/1.9780898719598. [30] J. Zabczyk, Mathematical Control Theory: An Introduction, Reprint of the 1995 edition, Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2008. doi: 10.1007/978-0-8176-4733-9.
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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