doi: 10.3934/eect.2021036
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Exact boundary null controllability for a coupled system of plate equations with variable coefficients

College of Science, Beijing Forestry University, Beijing, 100083, China

Received  January 2021 Revised  May 2021 Early access July 2021

This paper studies a coupled system of plate equations with variable coefficients, subject to the clamped boundary conditions. By the Riemannian geometry approach, the duality method, the multiplier technique and a compact perturbation method, we establish exact boundary null controllability of the system under verifiable assumptions.

Citation: Fengyan Yang. Exact boundary null controllability for a coupled system of plate equations with variable coefficients. Evolution Equations & Control Theory, doi: 10.3934/eect.2021036
References:
[1]

B. Allibert and S. Micu, Controllability of analytic functions for a wave equation coupled with a beam, Rev. Mat. Iberoamericana, 15 (1999), 547-592.  doi: 10.4171/RMI/265.  Google Scholar

[2]

G. Avalos, The exponential stability of a coupled hyerbolic/parabolic system arising in structural acoustics, Appl. Abstr. Anal., 1 (1996), 203-217.  doi: 10.1155/S1085337596000103.  Google Scholar

[3]

G. AvalosI. Lasiecka and R. Rebarber, Boundary controllability of a coupled wave/Kirchoff system, Systems Control Lett., 50 (2003), 331-341.  doi: 10.1016/S0167-6911(03)00179-8.  Google Scholar

[4]

K. Ammari and S. Nicaise, Stabilization of a transmission wave/plate equation, J. Differential Equations, 249 (2010), 707-727.  doi: 10.1016/j.jde.2010.03.007.  Google Scholar

[5]

G. AvalosI. Lasiecka and R. Triggiani, Heat-wave interaction in 2-3 dimensions: Optimal rational decay rate, J. Math. Anal. Appl., 437 (2016), 782-815.  doi: 10.1016/j.jmaa.2015.12.051.  Google Scholar

[6]

S. G. Chai and K. S. Liu, Boundary stabilization of the transmission of wave equations with variable coefficients, Chinese Ann. Math. Ser. A, 26 (2005), 605-612.   Google Scholar

[7]

S. Dolecki and D. L. Russell, A general theory of observation and control, SIAM J. Control Optim., 15 (1977), 185-220.  doi: 10.1137/0315015.  Google Scholar

[8]

L. Deng and Z. F. Zhang, Controllability for transmission wave/plate equations on Riemannian manifolds, Systems Control Lett., 91 (2016), 48-54.  doi: 10.1016/j.sysconle.2016.02.016.  Google Scholar

[9]

B. Z. Guo and Z. X. Zhang, Well-posedness and regularity for an Euler-Bernoulli plate with variable coefficients and boundary control and observation, Math. Control Signals Systems, 19 (2007), 337-360.  doi: 10.1007/s00498-007-0017-5.  Google Scholar

[10]

Y. X. Guo and P. F. Yao, Stabilization of Euler-Bernoulli plate equation with variable coefficients by nonlinear boundary feedback, J. Math. Anal. Appl., 317 (2006), 50-70.  doi: 10.1016/j.jmaa.2005.12.006.  Google Scholar

[11]

B. Z. Guo and Z. C. Shao, On well-posedness, regularity and exact controllability for problems of transmission of plate equation with variable coefficients, Quart. Appl. Math., 65 (2007), 705-736.  doi: 10.1090/S0033-569X-07-01069-9.  Google Scholar

[12]

F. Hassine, Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping, Discrete Contin. Dyn. Syst. Ser. B., 21 (2016), 1757-1774.  doi: 10.3934/dcdsb.2016021.  Google Scholar

[13]

M. A. Horn, Exact controllability of the Euler-Bernoulli plate via bending moments only on the space of optimal regularity, J. Math. Anal. Appl., 167 (1992), 557-581.  doi: 10.1016/0022-247X(92)90224-2.  Google Scholar

[14]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson, Paris, 1994.  Google Scholar

[15]

J. E. Lagnese and J. L. Lions, Modeling, Analysis and Control of Thin Plates, Masson, Paris, 1988.  Google Scholar

[16]

I. Lasiecka and R. Triggiani, Heat-structure interaction with viscoelastic damping: Analyticity with sharp analytic sector, exponential decay, fractional powers, Commun. Pure Appl. Anal., 15 (2016), 1515-1543.  doi: 10.3934/cpaa.2016001.  Google Scholar

[17]

I. Lasiecka and R. Triggiani, Exact controllability of the Euler-Bernoulli equation with controls in the Dirichlet and Neumann boundary conditions: A non-conservative case, SIAM J. Control Optim., 27 (1989), 330-373.  doi: 10.1137/0327018.  Google Scholar

[18]

I. Lasiecka and R. Triggiani, Exact controllability of the Euler-Bernoulli equation with boundary controls for displacement and moment, J. Math. Anal. Appl., 146 (1990), 1-33.  doi: 10.1016/0022-247X(90)90330-I.  Google Scholar

[19]

I. Lasiecka and J. T. Webster, Stabilization of a nonlinear flow-plate interaction via component-wise decomposition, Bull. Braz. Math. Soc. (N.S.)., 47 (2016), 489-506.  doi: 10.1007/s00574-016-0164-8.  Google Scholar

[20]

J. Li and S. G. Chai, Stabilization of the variable-coefficient structural acoustic model with curved middle surface and delay effects in the structural component, J. Math. Anal. Appl., 454 (2017), 510-532.  doi: 10.1016/j.jmaa.2017.05.001.  Google Scholar

[21]

T.-T. Li and B. P. Rao, Asymptotic controllability and asymptotic synchronization for a coupled system of wave equations with Dirichlet boundary controls, Asym. Anal., 86 (2014), 199-226.  doi: 10.3233/ASY-131193.  Google Scholar

[22]

T.-T. Li and B. P. Rao, Exact boundary controllability for a coupled system of wave equations with Neumann boundary controls, Chin. Ann. Math. Ser. B, 38 (2007), 473-488.  doi: 10.1007/s11401-017-1078-5.  Google Scholar

[23]

J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.  doi: 10.1137/1030001.  Google Scholar

[24]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Springer Verlag, New York, 1972.  Google Scholar

[25]

W. Liu and G. H. Williams, The exponential stability of the problem of transmission of the wave equation, Bull. Aust. Math. Soc., 57 (1998), 305-327.  doi: 10.1017/S0004972700031683.  Google Scholar

[26]

W. Liu and G. H. Williams, Exact controllability for problems of transmission of the plate equation with lower-order terms, Quart. Appl. Math., 58 (2000), 37-68.  doi: 10.1090/qam/1738557.  Google Scholar

[27]

S. Micu and E. Zuazua, Boundary controllability of a linear hybrid system arising in the control of noise, SIAM J. Control Optim., 35 (1997), 1614-1637.  doi: 10.1137/S0363012996297972.  Google Scholar

[28]

N. Ourada and R. Triggiani, Uniform stabilization of the EulerBernoulli equation with feedback operator only in the Neumann boundary condition, Differential Integral Equations, 4 (1991), 277-292.   Google Scholar

[29]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[30]

R. N. Pederson, On the unique continuation theorem for certain second and fourth order elliptic equations, Comm. Pure Appl. Math., 11 (1958), 67-80.  doi: 10.1002/cpa.3160110104.  Google Scholar

[31]

R. Szilard, Theories and Applications of Plate Analysis: Classical Numerical and Engineering Methods, John Wiley & Sons, Inc., Hoboken, New Jersey, 2004. doi: 10.1115/1.1849175.  Google Scholar

[32]

T. Shirota, A remark on the unique continuation theorem for certain fourth order elliptic equations, Proc. Japan Acad., 36 (1960), 571-573.   Google Scholar

[33]

D. Tataru, Boundary controllability for conservative PDEs, Appl. Math. Optim., 31 (1995), 257-295.  doi: 10.1007/BF01215993.  Google Scholar

[34]

E. Ventsel and T. Krauthammer, Thin Plates and Shells: Theory, Analysis, and Applications, Marcel Dekker, Inc., New York, 2001. doi: 10.1201/9780203908723.  Google Scholar

[35]

R. L. WenS. G. Chai and B. Z. Guo, Well-posedness and regularity of Euler-Bernoulli equation with variable coefficient and Dirichlet boundary control and collocated observation, Math. Methods Appl. Sci., 37 (2014), 2889-2905.  doi: 10.1002/mma.3028.  Google Scholar

[36]

P. F. Yao, Observability inequalities for the Euler-Bernoulli plate with variable coefficients, in Differential Geometric Methods in the Control of Partial Differential Equations, Contemp. Math., vol. 268, Amer. Math. Soc., Providence, RI, 2000,383–406. doi: 10.1090/conm/268/04320.  Google Scholar

[37]

P. F. Yao, Modeling and Control in Vibrational and Structual Dynamics: A Differential Geometric Approach, Chapman and Hall/CRC Applied Mathematics and Nonlinear Science Series, CRC Press, Boca Raton, FL, 2011. doi: 10.1201/b11042.  Google Scholar

[38]

F. Y. Yang, Exact controllability of the Euler-Bernoulli plate with variable coefficients and simply supported boundary condition, Electron. J. Diff. Eq., 257 (2016), 1-19.   Google Scholar

[39]

F. Y. YangB. Bin-MohsinG. Chen and P. F. Yao, Exact-approximate boundary controllability of the thermoelastic plate with a curved middle surface, J. Math. Anal. Appl., 451 (2017), 405-433.  doi: 10.1016/j.jmaa.2017.02.005.  Google Scholar

[40]

F. Y. YangP. F. Yao and G. Chen, Boundary controllability of structural acoustic systems with variable coefficients and curved walls, Math. Control Signals Syst., 30 (2018), 1-28.  doi: 10.1007/s00498-018-0211-7.  Google Scholar

[41]

W. Zhang and Z. F. Zhang, Stabilization of transmission coupled wave and Euler-Bernoulli equations on Riemannian manifolds by nonlinear feedbacks, J. Math. Anal. Appl., 422 (2015), 1504-1526.  doi: 10.1016/j.jmaa.2014.09.044.  Google Scholar

[42]

X. Zhang and E. Zuazua, Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction, Arch. Ration. Mech. Anal., 184 (2007), 49-120.  doi: 10.1007/s00205-006-0020-x.  Google Scholar

[43]

E. Zuazua, Controlabilite exacte d'un modele de plaques vibrantes en un temps arbitrairement petit, C. R. Acad. Sci. Paris Ser. I Math., 304 (1987), 173-176.   Google Scholar

show all references

References:
[1]

B. Allibert and S. Micu, Controllability of analytic functions for a wave equation coupled with a beam, Rev. Mat. Iberoamericana, 15 (1999), 547-592.  doi: 10.4171/RMI/265.  Google Scholar

[2]

G. Avalos, The exponential stability of a coupled hyerbolic/parabolic system arising in structural acoustics, Appl. Abstr. Anal., 1 (1996), 203-217.  doi: 10.1155/S1085337596000103.  Google Scholar

[3]

G. AvalosI. Lasiecka and R. Rebarber, Boundary controllability of a coupled wave/Kirchoff system, Systems Control Lett., 50 (2003), 331-341.  doi: 10.1016/S0167-6911(03)00179-8.  Google Scholar

[4]

K. Ammari and S. Nicaise, Stabilization of a transmission wave/plate equation, J. Differential Equations, 249 (2010), 707-727.  doi: 10.1016/j.jde.2010.03.007.  Google Scholar

[5]

G. AvalosI. Lasiecka and R. Triggiani, Heat-wave interaction in 2-3 dimensions: Optimal rational decay rate, J. Math. Anal. Appl., 437 (2016), 782-815.  doi: 10.1016/j.jmaa.2015.12.051.  Google Scholar

[6]

S. G. Chai and K. S. Liu, Boundary stabilization of the transmission of wave equations with variable coefficients, Chinese Ann. Math. Ser. A, 26 (2005), 605-612.   Google Scholar

[7]

S. Dolecki and D. L. Russell, A general theory of observation and control, SIAM J. Control Optim., 15 (1977), 185-220.  doi: 10.1137/0315015.  Google Scholar

[8]

L. Deng and Z. F. Zhang, Controllability for transmission wave/plate equations on Riemannian manifolds, Systems Control Lett., 91 (2016), 48-54.  doi: 10.1016/j.sysconle.2016.02.016.  Google Scholar

[9]

B. Z. Guo and Z. X. Zhang, Well-posedness and regularity for an Euler-Bernoulli plate with variable coefficients and boundary control and observation, Math. Control Signals Systems, 19 (2007), 337-360.  doi: 10.1007/s00498-007-0017-5.  Google Scholar

[10]

Y. X. Guo and P. F. Yao, Stabilization of Euler-Bernoulli plate equation with variable coefficients by nonlinear boundary feedback, J. Math. Anal. Appl., 317 (2006), 50-70.  doi: 10.1016/j.jmaa.2005.12.006.  Google Scholar

[11]

B. Z. Guo and Z. C. Shao, On well-posedness, regularity and exact controllability for problems of transmission of plate equation with variable coefficients, Quart. Appl. Math., 65 (2007), 705-736.  doi: 10.1090/S0033-569X-07-01069-9.  Google Scholar

[12]

F. Hassine, Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping, Discrete Contin. Dyn. Syst. Ser. B., 21 (2016), 1757-1774.  doi: 10.3934/dcdsb.2016021.  Google Scholar

[13]

M. A. Horn, Exact controllability of the Euler-Bernoulli plate via bending moments only on the space of optimal regularity, J. Math. Anal. Appl., 167 (1992), 557-581.  doi: 10.1016/0022-247X(92)90224-2.  Google Scholar

[14]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson, Paris, 1994.  Google Scholar

[15]

J. E. Lagnese and J. L. Lions, Modeling, Analysis and Control of Thin Plates, Masson, Paris, 1988.  Google Scholar

[16]

I. Lasiecka and R. Triggiani, Heat-structure interaction with viscoelastic damping: Analyticity with sharp analytic sector, exponential decay, fractional powers, Commun. Pure Appl. Anal., 15 (2016), 1515-1543.  doi: 10.3934/cpaa.2016001.  Google Scholar

[17]

I. Lasiecka and R. Triggiani, Exact controllability of the Euler-Bernoulli equation with controls in the Dirichlet and Neumann boundary conditions: A non-conservative case, SIAM J. Control Optim., 27 (1989), 330-373.  doi: 10.1137/0327018.  Google Scholar

[18]

I. Lasiecka and R. Triggiani, Exact controllability of the Euler-Bernoulli equation with boundary controls for displacement and moment, J. Math. Anal. Appl., 146 (1990), 1-33.  doi: 10.1016/0022-247X(90)90330-I.  Google Scholar

[19]

I. Lasiecka and J. T. Webster, Stabilization of a nonlinear flow-plate interaction via component-wise decomposition, Bull. Braz. Math. Soc. (N.S.)., 47 (2016), 489-506.  doi: 10.1007/s00574-016-0164-8.  Google Scholar

[20]

J. Li and S. G. Chai, Stabilization of the variable-coefficient structural acoustic model with curved middle surface and delay effects in the structural component, J. Math. Anal. Appl., 454 (2017), 510-532.  doi: 10.1016/j.jmaa.2017.05.001.  Google Scholar

[21]

T.-T. Li and B. P. Rao, Asymptotic controllability and asymptotic synchronization for a coupled system of wave equations with Dirichlet boundary controls, Asym. Anal., 86 (2014), 199-226.  doi: 10.3233/ASY-131193.  Google Scholar

[22]

T.-T. Li and B. P. Rao, Exact boundary controllability for a coupled system of wave equations with Neumann boundary controls, Chin. Ann. Math. Ser. B, 38 (2007), 473-488.  doi: 10.1007/s11401-017-1078-5.  Google Scholar

[23]

J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.  doi: 10.1137/1030001.  Google Scholar

[24]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Springer Verlag, New York, 1972.  Google Scholar

[25]

W. Liu and G. H. Williams, The exponential stability of the problem of transmission of the wave equation, Bull. Aust. Math. Soc., 57 (1998), 305-327.  doi: 10.1017/S0004972700031683.  Google Scholar

[26]

W. Liu and G. H. Williams, Exact controllability for problems of transmission of the plate equation with lower-order terms, Quart. Appl. Math., 58 (2000), 37-68.  doi: 10.1090/qam/1738557.  Google Scholar

[27]

S. Micu and E. Zuazua, Boundary controllability of a linear hybrid system arising in the control of noise, SIAM J. Control Optim., 35 (1997), 1614-1637.  doi: 10.1137/S0363012996297972.  Google Scholar

[28]

N. Ourada and R. Triggiani, Uniform stabilization of the EulerBernoulli equation with feedback operator only in the Neumann boundary condition, Differential Integral Equations, 4 (1991), 277-292.   Google Scholar

[29]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[30]

R. N. Pederson, On the unique continuation theorem for certain second and fourth order elliptic equations, Comm. Pure Appl. Math., 11 (1958), 67-80.  doi: 10.1002/cpa.3160110104.  Google Scholar

[31]

R. Szilard, Theories and Applications of Plate Analysis: Classical Numerical and Engineering Methods, John Wiley & Sons, Inc., Hoboken, New Jersey, 2004. doi: 10.1115/1.1849175.  Google Scholar

[32]

T. Shirota, A remark on the unique continuation theorem for certain fourth order elliptic equations, Proc. Japan Acad., 36 (1960), 571-573.   Google Scholar

[33]

D. Tataru, Boundary controllability for conservative PDEs, Appl. Math. Optim., 31 (1995), 257-295.  doi: 10.1007/BF01215993.  Google Scholar

[34]

E. Ventsel and T. Krauthammer, Thin Plates and Shells: Theory, Analysis, and Applications, Marcel Dekker, Inc., New York, 2001. doi: 10.1201/9780203908723.  Google Scholar

[35]

R. L. WenS. G. Chai and B. Z. Guo, Well-posedness and regularity of Euler-Bernoulli equation with variable coefficient and Dirichlet boundary control and collocated observation, Math. Methods Appl. Sci., 37 (2014), 2889-2905.  doi: 10.1002/mma.3028.  Google Scholar

[36]

P. F. Yao, Observability inequalities for the Euler-Bernoulli plate with variable coefficients, in Differential Geometric Methods in the Control of Partial Differential Equations, Contemp. Math., vol. 268, Amer. Math. Soc., Providence, RI, 2000,383–406. doi: 10.1090/conm/268/04320.  Google Scholar

[37]

P. F. Yao, Modeling and Control in Vibrational and Structual Dynamics: A Differential Geometric Approach, Chapman and Hall/CRC Applied Mathematics and Nonlinear Science Series, CRC Press, Boca Raton, FL, 2011. doi: 10.1201/b11042.  Google Scholar

[38]

F. Y. Yang, Exact controllability of the Euler-Bernoulli plate with variable coefficients and simply supported boundary condition, Electron. J. Diff. Eq., 257 (2016), 1-19.   Google Scholar

[39]

F. Y. YangB. Bin-MohsinG. Chen and P. F. Yao, Exact-approximate boundary controllability of the thermoelastic plate with a curved middle surface, J. Math. Anal. Appl., 451 (2017), 405-433.  doi: 10.1016/j.jmaa.2017.02.005.  Google Scholar

[40]

F. Y. YangP. F. Yao and G. Chen, Boundary controllability of structural acoustic systems with variable coefficients and curved walls, Math. Control Signals Syst., 30 (2018), 1-28.  doi: 10.1007/s00498-018-0211-7.  Google Scholar

[41]

W. Zhang and Z. F. Zhang, Stabilization of transmission coupled wave and Euler-Bernoulli equations on Riemannian manifolds by nonlinear feedbacks, J. Math. Anal. Appl., 422 (2015), 1504-1526.  doi: 10.1016/j.jmaa.2014.09.044.  Google Scholar

[42]

X. Zhang and E. Zuazua, Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction, Arch. Ration. Mech. Anal., 184 (2007), 49-120.  doi: 10.1007/s00205-006-0020-x.  Google Scholar

[43]

E. Zuazua, Controlabilite exacte d'un modele de plaques vibrantes en un temps arbitrairement petit, C. R. Acad. Sci. Paris Ser. I Math., 304 (1987), 173-176.   Google Scholar

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