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  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 & Management Optimization, 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.  Google Scholar

[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.  Google Scholar

[3] R. A. Adams, Sobolev Spaces, 1$^nd$ edition, Academic Press, New York, 1975.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

L. C. Evans, Partial Differential Equations, 2$^nd$ edition, American Mathematical Society, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[18]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Annals of Differential Equations, 1 (1985), 43–56.  Google Scholar

[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.  Google Scholar

[20]

J. E. Lagnese, Boundary Stabilization of Thin Plates, Philadelphia: SIAM J. Control Optim, 1989. doi: 10.1137/1.9781611970821.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

Ö. Morgül, Dynamic boundary control of a Timoshenko beam, Automatica, 28 (1992), 1255–1260. doi: 10.1016/0005-1098(92)90070-V.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

J. Prüss, On the spectrum of $C_0$-semigroups, Transactions of the American Mathematical Society, 284 (1984), 847–857. doi: 10.2307/1999112.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3] R. A. Adams, Sobolev Spaces, 1$^nd$ edition, Academic Press, New York, 1975.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

L. C. Evans, Partial Differential Equations, 2$^nd$ edition, American Mathematical Society, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[18]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Annals of Differential Equations, 1 (1985), 43–56.  Google Scholar

[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.  Google Scholar

[20]

J. E. Lagnese, Boundary Stabilization of Thin Plates, Philadelphia: SIAM J. Control Optim, 1989. doi: 10.1137/1.9781611970821.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

Ö. Morgül, Dynamic boundary control of a Timoshenko beam, Automatica, 28 (1992), 1255–1260. doi: 10.1016/0005-1098(92)90070-V.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

J. Prüss, On the spectrum of $C_0$-semigroups, Transactions of the American Mathematical Society, 284 (1984), 847–857. doi: 10.2307/1999112.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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