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March  2022, 18(2): 1009-1034. doi: 10.3934/jimo.2021006

## Stabilization of 2-d Mindlin-Timoshenko plates with localized acoustic boundary feedback

 1 Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang 310018, China 2 School of Mechanical Engineering & Mechanics, Ningbo University, Ningbo, Zhejiang 315211, China 3 Ningbo Artificial Intelligence Institute, Shanghai Jiao Tong University, Ningbo, Zhejiang 315000, China

* Corresponding author: Chunguo Zhang, Tehuan Chen

Received  September 2020 Revised  November 2020 Published  March 2022 Early access  December 2020

Fund Project: This research was supported by the National Key R & D Program of China (No. 2019YFB1705800), the National Natural Science Foundation of China under (No. 61973270), the Fundamental Research Funds for the Central Universities of China (No. 2018XZZX001-09) and Zhejiang Provincial Natural Science Foundation of China (No. LY21F030003)

In this paper, we investigate the well-posedness and the asymptotic stability of a two dimensional Mindlin-Timoshenko plate imposed the so-called acoustic control by a part of the boundary and a Dirichlet boundary condition on the remainder. We first establish the well-posedness results of our model based on the theory of linear operator semigroup and then prove that the system is not exponentially stable by using the frequency domain approach. Finally, we show that the system is polynomially stable with the aid of the exponential or polynomial stability of a system with standard damping acting on a part of the boundary.

Citation: Yubiao Liu, Chunguo Zhang, Tehuan Chen. Stabilization of 2-d Mindlin-Timoshenko plates with localized acoustic boundary feedback. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1009-1034. doi: 10.3934/jimo.2021006
##### References:
 [1] Z. Abbas and S. Nicaise, The Multidimensional wave equation with generalized acoustic boundary conditions I: Strong stability, SIAM J. Control Optim, 53 (2015), 2558–2581. doi: 10.1137/140971336. [2] Z. Abbas and S. Nicaise, The Multidimensional wave equation with generalized acoustic boundary conditions II: Polynomial stability, SIAM J. Control Optim, 53 (2015), 2582–2607. doi: 10.1137/140971348. [3] R. A. Adams, Sobolev Spaces, 1$^nd$ edition, Academic Press, New York, 1975. [4] H. Barucq, J. Diaz and V. Duprat, Long-term stability analsis of acoustic absorbing boundary conditions, Mathematical Models and Methods in Applied Sciences, 23 (2013), 2129–2154. doi: 10.1142/S0218202513500280. [5] J. T. Beale, Spectral properties of an acoustic boundary condition, Indiana University Mathematics Journal, 25 (1976), 895–917. doi: 10.1512/iumj.1976.25.25071. [6] A. Benaissa and A. Kasmi, Well-posedeness and energy decay of solutions to a bresse system with a boundary dissipation of fractional derivative type, Discrete & Continuous Dynamical Systems-Series-B, 23 (2018), 4361–4395. doi: 10.3934/dcdsb.2018168. [7] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Mathematische Annalen, 347 (2010), 455–478. doi: 10.1007/s00208-009-0439-0. [8] X. Cai, L. Liao and Y. Sun, Global strong solution to the initial-boundary value problem of a 2-D Kazhikhov-Smagulov type model, Discrete & Continuous Dynamical Systems-Series-S, 7 (2014), 917–923. doi: 10.3934/dcdss.2014.7.917. [9] G. Chen, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, Journal de Mathematiques et Pure Appliquees, 58 (1979), 249–273. [10] C. M. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Archive for Rational Mechanics and Analysis, 29 (1968), 241–271. doi: 10.1007/BF00276727. [11] M. G.-V. Dalsen, Exponential stabilization of magnetoelastic waves in a Mindlin-Timoshenko plate by localized internal damping, Zeitschrift für angewandte Mathematik und Physik, 66 (2015), 1751–1776. doi: 10.1007/s00033-015-0507-0. [12] M. S. de Queiroz, D.M. Dawson, M. Agarwal and F. Zhang, Adaptive nonlinear boundary control of a flexible link robot arm, IEEE Conference on Decision and Control, (2002). doi: 10.1109/CDC.1997.657642. [13] S.-R. Deng, B.-R. Lu, B.-Q. Dong, et al, Effective polarization control of metallic planar chiral metamaterials with complementary rosette pattern fabricated by nanoimprint lithography, Microelectronic Engineering, 87 (2013), 985–988. doi: 10.1016/j.mee.2009.11.123. [14] L. C. Evans, Partial Differential Equations, 2$^nd$ edition, American Mathematical Society, 1998. [15] L. Gearhart, Spectral theory for contraction semigroups on Hilbert space, Transactions of the American Mathematical Society, 236 (1978), 385–394. doi: 10.1090/S0002-9947-1978-0461206-1. [16] B.-Z. Guo and C.-Z. Xu, The stabilization of a one-dimensional wave equation by boundary feedback with non-collocated observation, IEEE Transactions on Automatic Control, 52 (2007), 371–377. doi: 10.1109/TAC.2006.890385. [17] J. Henry, J. Blondeau and D. Pines, Stability analysis for UAVs with a variable aspect ratio wing, AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 46 (2005), 18–21. [18] F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Annals of Differential Equations, 1 (1985), 43–56. [19] M. Krstic, D. Fontaine, P. V. Kokotovic and J. D. Paduano, Useful nonlinearities and global stabilization of bifurcations in a model of jet engine surge and stall, IEEE Transactions on Automatic Control, 43 (1998), 1739–1745. doi: 10.1109/9.736075. [20] J. E. Lagnese, Boundary Stabilization of Thin Plates, Philadelphia: SIAM J. Control Optim, 1989. doi: 10.1137/1.9781611970821. [21] J. E. Lagnese and J. L. Lions, Modelling, Analysis and Control of Thin Plates, 1$^{nd}$ edition, SIAM Journal on Control and Optimization, Philadelphia, 1989. doi: 10.1137/1.9781611970821. [22] C. Y. Lin, E. F. Crawley and J. Heeg, Open and closed-loop results of a strain-actuated active aeroelastic wing, Journal of Aircraft, 33 (2012), 987–994. doi: 10.2514/3.47045. [23] B. Liu and W. Littman, On the spectral properties and stabilization of acoustic flow, SIAM J. Appl. Math, 59 (1998), 17–34. doi: 10.1137/S0036139996314106. [24] K. Liu and Z. Liu, Exponential decay of energy of vibrating strings with local viscoelasticity, Z. Angew. Math. Phys, 53 (2002), 265–280. doi: 10.1007/s00033-002-8155-6. [25] S. A. Messaoudi and M. I. Mustafa, A general stability result in a memory-type Timoshenko system, Communications on Pure & Applied Analysis, 12 (2013), 957–972. doi: 10.3934/cpaa.2013.12.957. [26] Ö. Morgül, Dynamic boundary control of a Euler-Bernoulli beam, IEEE Transactions on Automatic Control, 37 (2002), 639–642. doi: 10.1109/9.135504. [27] Ö. Morgül, Dynamic boundary control of a Timoshenko beam, Automatica, 28 (1992), 1255–1260. doi: 10.1016/0005-1098(92)90070-V. [28] M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation, Mathematische Annalen, 305 (1996), 403–417. doi: 10.1007/BF01444231. [29] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, 2$^{nd}$ edition Springer-Verlag New York, 1983. doi: 10.1007/978-1-4612-5561-1. [30] J. Prüss, On the spectrum of $C_0$-semigroups, Transactions of the American Mathematical Society, 284 (1984), 847–857. doi: 10.2307/1999112. [31] J. E. M. Rivera1 and Y. Qin, Polynomial decay for the energy with an acoustic boundary condition, Applied Mathematics Letters, 16 (2003), 249–256. doi: 10.1016/S0893-9659(03)80039-3. [32] D. L. Russell and B. Y. Zhang, Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain, SIAM J. Control Optim, 31 (1993), 659–676. doi: 10.1137/0331030. [33] J.-H. Ryu, D.-S. Kwon, and B. Hannaford, Control of a flexible manipulator with noncollocated feedback: Time-domain passivity approach, IEEE Transactions on Robotics, 20 (2004), 776–780. doi: 10.1109/TRO.2004.829454. [34] H. D. F. Sare, On the stability of Mindlin-Timoshenko plates, Quarterly Journal of Mechanics & Applied Mathematics, 67 (2009), 249–263. doi: 10.1090/S0033-569X-09-01110-2. [35] A. Smyshlyaev, B.-Z. Guo and Miroslav Krstic, Arbitrary decay rate for Euler-Bernoulli beam by backstepping boundary feedback, IEEE Transactions on Automatic Control, 54 (2009), 1134–1140. doi: 10.1109/TAC.2009.2013038. [36] L. Tebou, Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the p-Laplacian, Discrete & Continuous Dynamical Systems-Series A, 32 (2012), 2315–2337. doi: 10.3934/dcds.2012.32.2315. [37] A. Wehbe, Rational energy decay rate for a wave equation with dynamical control, Applied Mathematics Letters, 16 (2003), 357–364. doi: 10.1016/S0893-9659(03)80057-5.

show all references

##### References:
 [1] Z. Abbas and S. Nicaise, The Multidimensional wave equation with generalized acoustic boundary conditions I: Strong stability, SIAM J. Control Optim, 53 (2015), 2558–2581. doi: 10.1137/140971336. [2] Z. Abbas and S. Nicaise, The Multidimensional wave equation with generalized acoustic boundary conditions II: Polynomial stability, SIAM J. Control Optim, 53 (2015), 2582–2607. doi: 10.1137/140971348. [3] R. A. Adams, Sobolev Spaces, 1$^nd$ edition, Academic Press, New York, 1975. [4] H. Barucq, J. Diaz and V. Duprat, Long-term stability analsis of acoustic absorbing boundary conditions, Mathematical Models and Methods in Applied Sciences, 23 (2013), 2129–2154. doi: 10.1142/S0218202513500280. [5] J. T. Beale, Spectral properties of an acoustic boundary condition, Indiana University Mathematics Journal, 25 (1976), 895–917. doi: 10.1512/iumj.1976.25.25071. [6] A. Benaissa and A. Kasmi, Well-posedeness and energy decay of solutions to a bresse system with a boundary dissipation of fractional derivative type, Discrete & Continuous Dynamical Systems-Series-B, 23 (2018), 4361–4395. doi: 10.3934/dcdsb.2018168. [7] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Mathematische Annalen, 347 (2010), 455–478. doi: 10.1007/s00208-009-0439-0. [8] X. Cai, L. Liao and Y. Sun, Global strong solution to the initial-boundary value problem of a 2-D Kazhikhov-Smagulov type model, Discrete & Continuous Dynamical Systems-Series-S, 7 (2014), 917–923. doi: 10.3934/dcdss.2014.7.917. [9] G. Chen, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, Journal de Mathematiques et Pure Appliquees, 58 (1979), 249–273. [10] C. M. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Archive for Rational Mechanics and Analysis, 29 (1968), 241–271. doi: 10.1007/BF00276727. [11] M. G.-V. Dalsen, Exponential stabilization of magnetoelastic waves in a Mindlin-Timoshenko plate by localized internal damping, Zeitschrift für angewandte Mathematik und Physik, 66 (2015), 1751–1776. doi: 10.1007/s00033-015-0507-0. [12] M. S. de Queiroz, D.M. Dawson, M. Agarwal and F. Zhang, Adaptive nonlinear boundary control of a flexible link robot arm, IEEE Conference on Decision and Control, (2002). doi: 10.1109/CDC.1997.657642. [13] S.-R. Deng, B.-R. Lu, B.-Q. Dong, et al, Effective polarization control of metallic planar chiral metamaterials with complementary rosette pattern fabricated by nanoimprint lithography, Microelectronic Engineering, 87 (2013), 985–988. doi: 10.1016/j.mee.2009.11.123. [14] L. C. Evans, Partial Differential Equations, 2$^nd$ edition, American Mathematical Society, 1998. [15] L. Gearhart, Spectral theory for contraction semigroups on Hilbert space, Transactions of the American Mathematical Society, 236 (1978), 385–394. doi: 10.1090/S0002-9947-1978-0461206-1. [16] B.-Z. Guo and C.-Z. Xu, The stabilization of a one-dimensional wave equation by boundary feedback with non-collocated observation, IEEE Transactions on Automatic Control, 52 (2007), 371–377. doi: 10.1109/TAC.2006.890385. [17] J. Henry, J. Blondeau and D. Pines, Stability analysis for UAVs with a variable aspect ratio wing, AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 46 (2005), 18–21. [18] F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Annals of Differential Equations, 1 (1985), 43–56. [19] M. Krstic, D. Fontaine, P. V. Kokotovic and J. D. Paduano, Useful nonlinearities and global stabilization of bifurcations in a model of jet engine surge and stall, IEEE Transactions on Automatic Control, 43 (1998), 1739–1745. doi: 10.1109/9.736075. [20] J. E. Lagnese, Boundary Stabilization of Thin Plates, Philadelphia: SIAM J. Control Optim, 1989. doi: 10.1137/1.9781611970821. [21] J. E. Lagnese and J. L. Lions, Modelling, Analysis and Control of Thin Plates, 1$^{nd}$ edition, SIAM Journal on Control and Optimization, Philadelphia, 1989. doi: 10.1137/1.9781611970821. [22] C. Y. Lin, E. F. Crawley and J. Heeg, Open and closed-loop results of a strain-actuated active aeroelastic wing, Journal of Aircraft, 33 (2012), 987–994. doi: 10.2514/3.47045. [23] B. Liu and W. Littman, On the spectral properties and stabilization of acoustic flow, SIAM J. Appl. Math, 59 (1998), 17–34. doi: 10.1137/S0036139996314106. [24] K. Liu and Z. Liu, Exponential decay of energy of vibrating strings with local viscoelasticity, Z. Angew. Math. Phys, 53 (2002), 265–280. doi: 10.1007/s00033-002-8155-6. [25] S. A. Messaoudi and M. I. Mustafa, A general stability result in a memory-type Timoshenko system, Communications on Pure & Applied Analysis, 12 (2013), 957–972. doi: 10.3934/cpaa.2013.12.957. [26] Ö. Morgül, Dynamic boundary control of a Euler-Bernoulli beam, IEEE Transactions on Automatic Control, 37 (2002), 639–642. doi: 10.1109/9.135504. [27] Ö. Morgül, Dynamic boundary control of a Timoshenko beam, Automatica, 28 (1992), 1255–1260. doi: 10.1016/0005-1098(92)90070-V. [28] M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation, Mathematische Annalen, 305 (1996), 403–417. doi: 10.1007/BF01444231. [29] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, 2$^{nd}$ edition Springer-Verlag New York, 1983. doi: 10.1007/978-1-4612-5561-1. [30] J. Prüss, On the spectrum of $C_0$-semigroups, Transactions of the American Mathematical Society, 284 (1984), 847–857. doi: 10.2307/1999112. [31] J. E. M. Rivera1 and Y. Qin, Polynomial decay for the energy with an acoustic boundary condition, Applied Mathematics Letters, 16 (2003), 249–256. doi: 10.1016/S0893-9659(03)80039-3. [32] D. L. Russell and B. Y. Zhang, Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain, SIAM J. Control Optim, 31 (1993), 659–676. doi: 10.1137/0331030. [33] J.-H. Ryu, D.-S. Kwon, and B. Hannaford, Control of a flexible manipulator with noncollocated feedback: Time-domain passivity approach, IEEE Transactions on Robotics, 20 (2004), 776–780. doi: 10.1109/TRO.2004.829454. [34] H. D. F. Sare, On the stability of Mindlin-Timoshenko plates, Quarterly Journal of Mechanics & Applied Mathematics, 67 (2009), 249–263. doi: 10.1090/S0033-569X-09-01110-2. [35] A. Smyshlyaev, B.-Z. Guo and Miroslav Krstic, Arbitrary decay rate for Euler-Bernoulli beam by backstepping boundary feedback, IEEE Transactions on Automatic Control, 54 (2009), 1134–1140. doi: 10.1109/TAC.2009.2013038. [36] L. Tebou, Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the p-Laplacian, Discrete & Continuous Dynamical Systems-Series A, 32 (2012), 2315–2337. doi: 10.3934/dcds.2012.32.2315. [37] A. Wehbe, Rational energy decay rate for a wave equation with dynamical control, Applied Mathematics Letters, 16 (2003), 357–364. doi: 10.1016/S0893-9659(03)80057-5.
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